- Email: [email protected]
Potential distribution, torsion problem and conformal mapping for a doubly-connected region with inner elliptic contour
compulers di SIru&m?S Vol. 31, No. 5, pp. 751-m Printed in Gnat Britain.
1989
POTENTIAL DISTRIBUTION, TORSION PROBLEM AND CONFORMAL MAPPING FOR A DOUBLY-CONNECTED REGION WITH INNER ELLIPTIC CONTOUR X. Y. LINT and R. L. CHEN$ t~pa~rnent
of Electrical Engin~~ng, P.O. Box 36, Jiangsu Institute of Technology, Zhenjiang, Jiangsu 212013, The People’s Republic of China and @kpartment of Computer Science, University of California, Santa Barbara, CA 93117, U.S.A. (Received IS September 1987)
Abstract-This paper deals with three boundary value problems of the Laplace equation: (1) potential distribution of a doubly-connected region with inner circular or elliptic boundary-a problem in electrostatics; (2) Saint-Venant torsion of a bar with same region-a problem in elasticity; (3) conformal mapping of the same region into a ring region-a problem incomplexvariable function. it is proved here that the first is a Dirichlet oroblem while the second and third are modified Dir&let nroblems of Ladace’s equation, and a modified birichlet problem can be easily reduced to two Dirichlet problems. In aider to solve the Dirichlet problems, an eigenexpansion form is proposed. The eigenexpansion form always satisfies the Laplace equation and a particular condition along the inner boundary. The undetermined coefficients in the eigenexpansion form are determined by the use of the variational principle. Several numerical examples and calculated results are given.
1. INTRODUCI’ION
In many electric devices the capacitors are important parts of the whole set. Therefore, the knowledge of potential distribution and capacitance of capacitors becomes a basic problem in electrostatics 111.On the other hand, in applied mathematics the 2-D problem of determining the electrical potential can be reduced to a Dirichlet boundary value problem of Laplace’s equation. It is still significant at present to investigate the potential dist~bution of capacitors with some complex boundaries though the theory of electrostatics field is well known to us. The potential distribution and capacitance of capacitors with an inner elliptic contour are investigated in this paper. Firstly, we can easily derive an eigenexpansion form in which every term satisfies Laplace’s equation and inner boundary value condition. Secondly, the coefficients in eigenexpansion form can be obtained by the use of the variational principle. Finally, numerical examples and calculated results are given. The solution of the Saint-Venant torsion problem of a doubly-connected region with inner circular or elliptic boundary is more complicated in elasticity [2]. This paper takes the conjugate harmonic function (i.e. conjugate to the warping harmonic function in torsion problem) to be an unknown function, and the boundary value problem of torsion becomes the modified Dirichlet problem which can be further reduced to two Dirichlet problems mentioned above. The solution of the Dirichlet problem can be found out in a similar way. The single-valued condition of the warping function is satisfied by suitable
superposition of the solutions of the two Dirichlet problems. The equivalence between the conformal mapping problem and the Dirichlet problem of Laplace’s equation is well known to us [3]. Nevertheless, it is not easy to convert the mapping problem to a Dirichlet problem of the Laplace equation. It is also proved that the conformal mapping of the doubly-innate region with some inner and outer boundaries to the ring shape region can be reduced to a modified Dirichlet problem. Obviously, its solution is also obtainable in the same way. One important parameter in the conformal mapping is the ratio of the outer radius to the inner radius of the ring region. After the modified Dirichlet problem is solved, this parameter can also be calculated.
2. POTENTIAL DISTRIBUTION AND CAPCITANCE OF CAPACITORS WITH INNER ELLIPTIC BOUNDARY It is well known that the governing equation for the potential distribution of a 2-D capacitor, as shown in Fig. 1, can be reduced to the following boundary value problem of Laplace’s equation [ 11: VU/(x,
y) = 0
(la) (lb)
WI,, =O WI& = i7
(in this problem si = 1)
(lc)
where L, denotes the inner elliptic boundary. flere, it is assumed that the outer boundary Lt of the 751
X. Y. LIN and R. L.
152
CHEN
If the capacitor has two symmetric axes, as shown in Fig. 1, the linear combination
2b
(9)
Ll
always satisfies eqn (7). Letting
Fig. 1. A plane capacitor with inner elliptic contour.
capacitor can be arbitrary. Obviously, the capacitance of a capacitor can be expressed as follows
c
=d
(10)
sss, L2
(2)
an
where 6 is the permittivity of material and I is the length of the capacitor. Now, we want to find the functions U(x, JJ) which satisfy the conditions (la) and (lb). In order to find these functions, it is assumed that the function CJ(x,y) is the real part of an analytic function a(z). i.e. U(x,y) = Re @p(z)= (e(z) + @(z))/2.
from eqn (9) we have U(x, y) =
c X,U’k’(x,y).
k=l
(11)
Clearly, each term in eqn (11) satisfies conditions (la) and (1b) automatically. From the variational principle we know that, in the condition that the function U(x, y) satisfies (1b), the stationary value of the following functional [2]
(3)
In this case, the condition (lb) becomes (4)
wz)+@i(z))IL,=o.
(12)
In addition, the following conformal mapping u-cl 2
z ++~+--
i
or
1; = n(z)
(5)
maps the elliptic contour in the z-plane (z = x + iy) into a unit circle in the i-plane (c = < + ir]). After letting
the condition
leads to the conditions (la) and (1~). If U(x, y) also satisfies condition (la), from the following equation
then the functional
l7 shown by eqn (12) becomes
(lb) or (4) becomes
cp(a)+ v(a) = 0
(7)
(a = e”, points on the unit circle). It is easy to prove
that the functions
This is to say, if the function U(x, y) satisfies conditions (la) and (lb), the stationary value of the function l7 shown by eqn (14) leads to conditions (lc). Substituting eqn (11) into (14) and letting
anjax,=o, cpok-‘)(~)=i(Jk+~-k) @2k’([)=(jk-[-k) satisfy eqn (7).
k=l,2,... k=l,2,...
j=1,2,...M
(15)
we obtain (8) f cjkxk=Dj, k-l
j=l,&...hf
(16)
Boundary value problems of the Laplace equation
753
is as follows
where
c,, = c,, =
s
auu)
-
L2
an
cp([) = 1.30 log [ + 0.7016 x 10-9(c2 - c-*)
U’k’d.9
’
+ 0.6015 x 10-z([4 - c -‘) + 0.1202 x lo-‘(l6 -l-6)
(17a)
j,k=l,2,...M
+ 0.9350 x lo-S([8 - 1;-8) Dj =
s
au(j) -Uds,
L1 an
j=l,2
,...
M.
+ 0.2226 x lo-‘(l’* - [-I*)
Finally, eqn (16) is the governing equation for determining the coefficients Xk in eqn (9). Obviously, in the process of numerical integration in eqns (17a) and (17b) the values UCk)and au(j)~an at the discrete points on the outer boundary can be calculated as follows
au z
+0.5645 x lo-9([i0-[-‘o)
(17b)
+ 0.6250 x 10-“([‘4 - [ -14).
(20)
From the symmetric condition, the coefficients before ([2k - [ -2k), k = 1,3,5,7, should be equal to zero. The presented small values of these coefficients represent the round error in computation. In the same condition (M = 8, a = c, a/b = 0.5), the calculated potentials on one eighth of the region are expressed as follows U(x, Y) =_M& e),
-
ig
=
(21)
W(z) = cp’(~)/o’(~) where
g
=g
cos(n, x) +
if!! cos(n, y). ay
0 = arctan (y/x),
V-9
After the coefficients X, (k = 1,2, . . . M) are calculated from eqn (16), the potential distribution and capacitance of a capacitor with an inner elliptic contour can be evaluated. For explaining the above derivation, a numerical example and calculated results are presented. In the case of taking M = 8 in eqn (9), the calculated capacitance for the capacitor shown in Fig. 1 can be expressed as follows:
C =_tXalb,c/a)4
cos(Q)/(b - a cos(8)).
e = (x -a
(19)
The calculated X(6, e) values are listed in Table 2.
3.
SOLUTION OF THE SAINT-VENANT TORSION PROBLEM OF BAR WITH INNER ELLIPTIC BOUNDARY
It is well known that, if one takes the conjugate harmonic function $(x, y) (conjugate to the warping function, rp(x, y) in the torsion problem) to be an unknown function, the governing equation of the torsion problem becomes [2]
where c is the permittivity of material and 1 is the length of capacitor. The calculatedf, values are listed in Table 1. In addition, in the case of taking M = 8 in eqn (16), a = c and a/b = 0.5, the obtained numerical solution
V2$(x,y)=0
(23a)
ti~L,=(x*+Y*w+K
ti I L2 = (x2 +
W-4
Y2)/2.
(23~)
Table l.f, values in eqn (19) cla 0.5 1.0
0.1 2.357 2.642
(22)
0.2
0.3
0.4
a/b 0.5
0.6
0.7
0.8
0.9
3.185 3.729
4.010 4.910
4.919 6.334
5.981 8.173
7.295 10.72
9.048 14.57
11.69 21.31
16.96 37.18
754
LIN and R. L. f&EN
X. Y.
Table 2. A(@,e) values in eqn (21) e 0 (degrees)
0
0 5 IO 15
0
0 0 0 0 0 0 0 0 0
20 25 30 : 45
l/9 0.142 0.143 0.145 0.149 0.155 0.163 0.173 0.187 0.206 0.231
219 0.272 0.273 0.277 0.283 0.293 0.307 0.325 0.349 0.381 0.422
319 0.391 0.393 0.398 0.406 0.419 0.436 0.459 0.489 0.529 0.580
419 0.503 0.505 0.510 0.520 0.534 0.553 0.578 0.611 0.655 0.711
The single-valued condition of the warping function leads to the following equation [2]
519 0.609 0.611 0.616 0.626 0.640 0.659 0.684 0.717 0.760 0.815
Obviously, obtains
619 0.711 0.712 0.717 0.726 0.739 0.756 0.779 0.809 0.847 0.896
719 0.809 0.810 0.814 0.821 0.831 0.845 0.863 0.886 0.916 0.954
substituting
K =
where L’ is any closed path between the contours L, and L2 in Fig. 1, and n means the normal with respect to path L’. The constant Kin eqn (23b) can be determined by the use of condition (24). This is a well known modified Dirichlet problem of Dirichlet’s equation in complex variable function theory [3]. Note the property that the coordinates of the points (x, y) on the inner elliptic boundary satisfy
0f *+ 0f 2=1,
(25)
and let $(x,Y)=-&+$&~ + W* Y) + KU - WG Y)),
VI,=
(24), one
z*/z,
From the above derivations we see that, after the two Dirichlet problems shown by eqns (27) and (28) are solved, then the value of K can be obtained from (29). Finally, the conjugate function *(x, y) is obtainable. It is also obvious that the Dirichlet problems shown by eqns (27) and (28) are just the same as mentioned in eqn (l), and their solutions can be easily obtained. In addition, the torsional rigidity can be expressed as [2, P*471
(26) J=
(2$ -x2-y2)dx
dy +2KA,
=o, 1.
(27) J =&r/b,
VZT(x, y) = 0,
c/~)(2b)~.
(31)
Several calculated results are listed in Table 3. In the case of CI= c and a/b = 0.5, the calculated shear stresses along the boundary of one eighth of the section are shown in Fig. 2.
T/L, =0
Table 3. j&r/b, c/a) values [see Fig. 1 and eqn (31)] c/a
0.5 1.0
0.1
(30)
where R denotes the region occupied by the section and A = mc is the area of the ellipse (Fig. 1). In the case of M = 8, the calculated torsional rigidities can be expressed by
y) = 0, I$,
eqn (26) into
1.00 1.00 1.00 1.00 1.00 1.00 l.00 1.00 1.00 1.00
(29)
then we can separate the problem shown by eqn (23) into two Dirichlet problems as follows W(x,
1
819 0.905 0.906 0.908 0.912 0.918 0.926 0.936 0.950 0.967 0.987
0.2
0.3
0.4
alb 0.5
0.6
0.7
0.8
0.9
0.14160 0.14154 0.14126 0.14046 0.13868 0.13522 0.12892 0.11789 0.09772 0.14159 0.14144 0.14081 0.13908 0.13542 0.12866 0.11726 0.09901 0.07001
155
Boundary value problems of the Laplace equation
Using the following properties log 1W(z) I = log 1 = 0, 10g~z~=10g1=0,
ZEL,
log 1W(z)/ = log p = K log IZI =flog(x2
ZEL, (35)
ZEL, ZEL,
+y*)
we see that the harmonic function satisfy the following equation
(36)
P(x, y) should
VP(x, y) = 0
P(W)lL,=o Fig. 2. The calculated shear stresses 5 along the boundary of one-eighth of the section; T = (the value shown in figure) x M/b’; M = the applied torque.
4.
CONFORMAL MAPPING DOUBLY-CONNEaED
PROBLEM REGION
P(x,y)l,, = K -flog(x*+y’).
(37)
This is also a modified Dirichlet problem of Laplace’s equation in complex variable function theory [3]. In addition, after letting
OF
It is assumed that the following mapping function
W, Y) = KU, Y) - W, Y)
r = W(z)
the modified Dirichlet problem shown by eqn (37) can be separated into the following two Dirichlet problems
(32)
maps the doubly-connected region with inner circular boundary shown in Fig. 3a into the ring-shape region in Fig. 3b. From general theory of the conformal mapping we know that the value p in Fig. 3b representing the ratio of radii of two circles can not be arbitrary. Meanwhile, the value p will be obtained in the process of solving the modified Dirichlet problem. Obviously, the following function W)
= log(W)lr)
= P&y)
+ iQ(x,y)
fYX,Y) = lois 1W(z)1 -log LP
ty
z-
IzI. t
VV(x,y)=O
VI,,=o VI&= 1
(39)
VT(x, y) = 0
TI,, =O
(33)
is analytic on the region R. In other words, the real and imaginary parts of the function G(z) are some harmonic functions satisfying the single-valued condition. In addition, from (33) the harmonic function P(x,v) will be
(38)
TILl=flog(x*+y*).
(40)
Obviously, using eqns (24) and (38), one obtains K = Z2/Z,
(34) ,_
I, =
(41)
In the case of a = c and M = 10, the ratio of the outer radius to the inner radius of the ring-shape region can be expressed as
k--b+
P
I
Fig. 3. A doubly-connected region with inner circular boundary and its mapping.
Several calculated Table 4.
=f,(alb).
numerical
(42)
results are listed in
X. Y. LIN and R. L. CHEN
756
Table 4. f&z/b) values in eqn (42) e/b 0.1
f4 l/f,
10.79 0.0927
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
5.394 0.1854
0.3596 0.2781
2.691 0.3708
2.158 0.4636
1.791 0.5564
1.539 0.6498
1.343 0.7446
1.184 0.8445
In the case of a = c, a/b = 0.5 and M = 10, the obtained mapping function will be W(z) = zetiL’Z),
(43)
where
REFERENCES
1. R. K. Wangness, Electromagnetic Fielrfs. John Wiley, New York (1979).
2. S. Sokolnikoff, 3.
IL(z) = 0.4625 x IO-*(z4 - z-~) + 0.7283 x lo-‘(zs
4.
-z-*)
5.
+ 0.2152 x lo-‘(2’2 - -12) + 0.5147 x to-‘O(z’6 -z-l”).
6. (44)
Mathematical
Theory
of
Elasticity.
McGraw-Hill, New York (1956). S. G. Mikhlin, Integral Equations. Pergamon, London (1957). N. I. Muskhelishvili, Singular Integral equations. Noordhoff, Groningen (1953). R. Walter, Real and Complex Analyses. McGraw-Hill, New York (1974). Y. H. Chen, A method for conformal mapping of a two-connected region onto an annulus. Appl. Math. Me&
4(6), 853-860 (1983).
1989
POTENTIAL DISTRIBUTION, TORSION PROBLEM AND CONFORMAL MAPPING FOR A DOUBLY-CONNECTED REGION WITH INNER ELLIPTIC CONTOUR X. Y. LINT and R. L. CHEN$ t~pa~rnent
of Electrical Engin~~ng, P.O. Box 36, Jiangsu Institute of Technology, Zhenjiang, Jiangsu 212013, The People’s Republic of China and @kpartment of Computer Science, University of California, Santa Barbara, CA 93117, U.S.A. (Received IS September 1987)
Abstract-This paper deals with three boundary value problems of the Laplace equation: (1) potential distribution of a doubly-connected region with inner circular or elliptic boundary-a problem in electrostatics; (2) Saint-Venant torsion of a bar with same region-a problem in elasticity; (3) conformal mapping of the same region into a ring region-a problem incomplexvariable function. it is proved here that the first is a Dirichlet oroblem while the second and third are modified Dir&let nroblems of Ladace’s equation, and a modified birichlet problem can be easily reduced to two Dirichlet problems. In aider to solve the Dirichlet problems, an eigenexpansion form is proposed. The eigenexpansion form always satisfies the Laplace equation and a particular condition along the inner boundary. The undetermined coefficients in the eigenexpansion form are determined by the use of the variational principle. Several numerical examples and calculated results are given.
1. INTRODUCI’ION
In many electric devices the capacitors are important parts of the whole set. Therefore, the knowledge of potential distribution and capacitance of capacitors becomes a basic problem in electrostatics 111.On the other hand, in applied mathematics the 2-D problem of determining the electrical potential can be reduced to a Dirichlet boundary value problem of Laplace’s equation. It is still significant at present to investigate the potential dist~bution of capacitors with some complex boundaries though the theory of electrostatics field is well known to us. The potential distribution and capacitance of capacitors with an inner elliptic contour are investigated in this paper. Firstly, we can easily derive an eigenexpansion form in which every term satisfies Laplace’s equation and inner boundary value condition. Secondly, the coefficients in eigenexpansion form can be obtained by the use of the variational principle. Finally, numerical examples and calculated results are given. The solution of the Saint-Venant torsion problem of a doubly-connected region with inner circular or elliptic boundary is more complicated in elasticity [2]. This paper takes the conjugate harmonic function (i.e. conjugate to the warping harmonic function in torsion problem) to be an unknown function, and the boundary value problem of torsion becomes the modified Dirichlet problem which can be further reduced to two Dirichlet problems mentioned above. The solution of the Dirichlet problem can be found out in a similar way. The single-valued condition of the warping function is satisfied by suitable
superposition of the solutions of the two Dirichlet problems. The equivalence between the conformal mapping problem and the Dirichlet problem of Laplace’s equation is well known to us [3]. Nevertheless, it is not easy to convert the mapping problem to a Dirichlet problem of the Laplace equation. It is also proved that the conformal mapping of the doubly-innate region with some inner and outer boundaries to the ring shape region can be reduced to a modified Dirichlet problem. Obviously, its solution is also obtainable in the same way. One important parameter in the conformal mapping is the ratio of the outer radius to the inner radius of the ring region. After the modified Dirichlet problem is solved, this parameter can also be calculated.
2. POTENTIAL DISTRIBUTION AND CAPCITANCE OF CAPACITORS WITH INNER ELLIPTIC BOUNDARY It is well known that the governing equation for the potential distribution of a 2-D capacitor, as shown in Fig. 1, can be reduced to the following boundary value problem of Laplace’s equation [ 11: VU/(x,
y) = 0
(la) (lb)
WI,, =O WI& = i7
(in this problem si = 1)
(lc)
where L, denotes the inner elliptic boundary. flere, it is assumed that the outer boundary Lt of the 751
X. Y. LIN and R. L.
152
CHEN
If the capacitor has two symmetric axes, as shown in Fig. 1, the linear combination
2b
(9)
Ll
always satisfies eqn (7). Letting
Fig. 1. A plane capacitor with inner elliptic contour.
capacitor can be arbitrary. Obviously, the capacitance of a capacitor can be expressed as follows
c
=d
(10)
sss, L2
(2)
an
where 6 is the permittivity of material and I is the length of the capacitor. Now, we want to find the functions U(x, JJ) which satisfy the conditions (la) and (lb). In order to find these functions, it is assumed that the function CJ(x,y) is the real part of an analytic function a(z). i.e. U(x,y) = Re @p(z)= (e(z) + @(z))/2.
from eqn (9) we have U(x, y) =
c X,U’k’(x,y).
k=l
(11)
Clearly, each term in eqn (11) satisfies conditions (la) and (1b) automatically. From the variational principle we know that, in the condition that the function U(x, y) satisfies (1b), the stationary value of the following functional [2]
(3)
In this case, the condition (lb) becomes (4)
wz)+@i(z))IL,=o.
(12)
In addition, the following conformal mapping u-cl 2
z ++~+--
i
or
1; = n(z)
(5)
maps the elliptic contour in the z-plane (z = x + iy) into a unit circle in the i-plane (c = < + ir]). After letting
the condition
leads to the conditions (la) and (1~). If U(x, y) also satisfies condition (la), from the following equation
then the functional
l7 shown by eqn (12) becomes
(lb) or (4) becomes
cp(a)+ v(a) = 0
(7)
(a = e”, points on the unit circle). It is easy to prove
that the functions
This is to say, if the function U(x, y) satisfies conditions (la) and (lb), the stationary value of the function l7 shown by eqn (14) leads to conditions (lc). Substituting eqn (11) into (14) and letting
anjax,=o, cpok-‘)(~)=i(Jk+~-k) @2k’([)=(jk-[-k) satisfy eqn (7).
k=l,2,... k=l,2,...
j=1,2,...M
(15)
we obtain (8) f cjkxk=Dj, k-l
j=l,&...hf
(16)
Boundary value problems of the Laplace equation
753
is as follows
where
c,, = c,, =
s
auu)
-
L2
an
cp([) = 1.30 log [ + 0.7016 x 10-9(c2 - c-*)
U’k’d.9
’
+ 0.6015 x 10-z([4 - c -‘) + 0.1202 x lo-‘(l6 -l-6)
(17a)
j,k=l,2,...M
+ 0.9350 x lo-S([8 - 1;-8) Dj =
s
au(j) -Uds,
L1 an
j=l,2
,...
M.
+ 0.2226 x lo-‘(l’* - [-I*)
Finally, eqn (16) is the governing equation for determining the coefficients Xk in eqn (9). Obviously, in the process of numerical integration in eqns (17a) and (17b) the values UCk)and au(j)~an at the discrete points on the outer boundary can be calculated as follows
au z
+0.5645 x lo-9([i0-[-‘o)
(17b)
+ 0.6250 x 10-“([‘4 - [ -14).
(20)
From the symmetric condition, the coefficients before ([2k - [ -2k), k = 1,3,5,7, should be equal to zero. The presented small values of these coefficients represent the round error in computation. In the same condition (M = 8, a = c, a/b = 0.5), the calculated potentials on one eighth of the region are expressed as follows U(x, Y) =_M& e),
-
ig
=
(21)
W(z) = cp’(~)/o’(~) where
g
=g
cos(n, x) +
if!! cos(n, y). ay
0 = arctan (y/x),
V-9
After the coefficients X, (k = 1,2, . . . M) are calculated from eqn (16), the potential distribution and capacitance of a capacitor with an inner elliptic contour can be evaluated. For explaining the above derivation, a numerical example and calculated results are presented. In the case of taking M = 8 in eqn (9), the calculated capacitance for the capacitor shown in Fig. 1 can be expressed as follows:
C =_tXalb,c/a)4
cos(Q)/(b - a cos(8)).
e = (x -a
(19)
The calculated X(6, e) values are listed in Table 2.
3.
SOLUTION OF THE SAINT-VENANT TORSION PROBLEM OF BAR WITH INNER ELLIPTIC BOUNDARY
It is well known that, if one takes the conjugate harmonic function $(x, y) (conjugate to the warping function, rp(x, y) in the torsion problem) to be an unknown function, the governing equation of the torsion problem becomes [2]
where c is the permittivity of material and 1 is the length of capacitor. The calculatedf, values are listed in Table 1. In addition, in the case of taking M = 8 in eqn (16), a = c and a/b = 0.5, the obtained numerical solution
V2$(x,y)=0
(23a)
ti~L,=(x*+Y*w+K
ti I L2 = (x2 +
W-4
Y2)/2.
(23~)
Table l.f, values in eqn (19) cla 0.5 1.0
0.1 2.357 2.642
(22)
0.2
0.3
0.4
a/b 0.5
0.6
0.7
0.8
0.9
3.185 3.729
4.010 4.910
4.919 6.334
5.981 8.173
7.295 10.72
9.048 14.57
11.69 21.31
16.96 37.18
754
LIN and R. L. f&EN
X. Y.
Table 2. A(@,e) values in eqn (21) e 0 (degrees)
0
0 5 IO 15
0
0 0 0 0 0 0 0 0 0
20 25 30 : 45
l/9 0.142 0.143 0.145 0.149 0.155 0.163 0.173 0.187 0.206 0.231
219 0.272 0.273 0.277 0.283 0.293 0.307 0.325 0.349 0.381 0.422
319 0.391 0.393 0.398 0.406 0.419 0.436 0.459 0.489 0.529 0.580
419 0.503 0.505 0.510 0.520 0.534 0.553 0.578 0.611 0.655 0.711
The single-valued condition of the warping function leads to the following equation [2]
519 0.609 0.611 0.616 0.626 0.640 0.659 0.684 0.717 0.760 0.815
Obviously, obtains
619 0.711 0.712 0.717 0.726 0.739 0.756 0.779 0.809 0.847 0.896
719 0.809 0.810 0.814 0.821 0.831 0.845 0.863 0.886 0.916 0.954
substituting
K =
where L’ is any closed path between the contours L, and L2 in Fig. 1, and n means the normal with respect to path L’. The constant Kin eqn (23b) can be determined by the use of condition (24). This is a well known modified Dirichlet problem of Dirichlet’s equation in complex variable function theory [3]. Note the property that the coordinates of the points (x, y) on the inner elliptic boundary satisfy
0f *+ 0f 2=1,
(25)
and let $(x,Y)=-&+$&~ + W* Y) + KU - WG Y)),
VI,=
(24), one
z*/z,
From the above derivations we see that, after the two Dirichlet problems shown by eqns (27) and (28) are solved, then the value of K can be obtained from (29). Finally, the conjugate function *(x, y) is obtainable. It is also obvious that the Dirichlet problems shown by eqns (27) and (28) are just the same as mentioned in eqn (l), and their solutions can be easily obtained. In addition, the torsional rigidity can be expressed as [2, P*471
(26) J=
(2$ -x2-y2)dx
dy +2KA,
=o, 1.
(27) J =&r/b,
VZT(x, y) = 0,
c/~)(2b)~.
(31)
Several calculated results are listed in Table 3. In the case of CI= c and a/b = 0.5, the calculated shear stresses along the boundary of one eighth of the section are shown in Fig. 2.
T/L, =0
Table 3. j&r/b, c/a) values [see Fig. 1 and eqn (31)] c/a
0.5 1.0
0.1
(30)
where R denotes the region occupied by the section and A = mc is the area of the ellipse (Fig. 1). In the case of M = 8, the calculated torsional rigidities can be expressed by
y) = 0, I$,
eqn (26) into
1.00 1.00 1.00 1.00 1.00 1.00 l.00 1.00 1.00 1.00
(29)
then we can separate the problem shown by eqn (23) into two Dirichlet problems as follows W(x,
1
819 0.905 0.906 0.908 0.912 0.918 0.926 0.936 0.950 0.967 0.987
0.2
0.3
0.4
alb 0.5
0.6
0.7
0.8
0.9
0.14160 0.14154 0.14126 0.14046 0.13868 0.13522 0.12892 0.11789 0.09772 0.14159 0.14144 0.14081 0.13908 0.13542 0.12866 0.11726 0.09901 0.07001
155
Boundary value problems of the Laplace equation
Using the following properties log 1W(z) I = log 1 = 0, 10g~z~=10g1=0,
ZEL,
log 1W(z)/ = log p = K log IZI =flog(x2
ZEL, (35)
ZEL, ZEL,
+y*)
we see that the harmonic function satisfy the following equation
(36)
P(x, y) should
VP(x, y) = 0
P(W)lL,=o Fig. 2. The calculated shear stresses 5 along the boundary of one-eighth of the section; T = (the value shown in figure) x M/b’; M = the applied torque.
4.
CONFORMAL MAPPING DOUBLY-CONNEaED
PROBLEM REGION
P(x,y)l,, = K -flog(x*+y’).
(37)
This is also a modified Dirichlet problem of Laplace’s equation in complex variable function theory [3]. In addition, after letting
OF
It is assumed that the following mapping function
W, Y) = KU, Y) - W, Y)
r = W(z)
the modified Dirichlet problem shown by eqn (37) can be separated into the following two Dirichlet problems
(32)
maps the doubly-connected region with inner circular boundary shown in Fig. 3a into the ring-shape region in Fig. 3b. From general theory of the conformal mapping we know that the value p in Fig. 3b representing the ratio of radii of two circles can not be arbitrary. Meanwhile, the value p will be obtained in the process of solving the modified Dirichlet problem. Obviously, the following function W)
= log(W)lr)
= P&y)
+ iQ(x,y)
fYX,Y) = lois 1W(z)1 -log LP
ty
z-
IzI. t
VV(x,y)=O
VI,,=o VI&= 1
(39)
VT(x, y) = 0
TI,, =O
(33)
is analytic on the region R. In other words, the real and imaginary parts of the function G(z) are some harmonic functions satisfying the single-valued condition. In addition, from (33) the harmonic function P(x,v) will be
(38)
TILl=flog(x*+y*).
(40)
Obviously, using eqns (24) and (38), one obtains K = Z2/Z,
(34) ,_
I, =
(41)
In the case of a = c and M = 10, the ratio of the outer radius to the inner radius of the ring-shape region can be expressed as
k--b+
P
I
Fig. 3. A doubly-connected region with inner circular boundary and its mapping.
Several calculated Table 4.
=f,(alb).
numerical
(42)
results are listed in
X. Y. LIN and R. L. CHEN
756
Table 4. f&z/b) values in eqn (42) e/b 0.1
f4 l/f,
10.79 0.0927
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
5.394 0.1854
0.3596 0.2781
2.691 0.3708
2.158 0.4636
1.791 0.5564
1.539 0.6498
1.343 0.7446
1.184 0.8445
In the case of a = c, a/b = 0.5 and M = 10, the obtained mapping function will be W(z) = zetiL’Z),
(43)
where
REFERENCES
1. R. K. Wangness, Electromagnetic Fielrfs. John Wiley, New York (1979).
2. S. Sokolnikoff, 3.
IL(z) = 0.4625 x IO-*(z4 - z-~) + 0.7283 x lo-‘(zs
4.
-z-*)
5.
+ 0.2152 x lo-‘(2’2 - -12) + 0.5147 x to-‘O(z’6 -z-l”).
6. (44)
Mathematical
Theory
of
Elasticity.
McGraw-Hill, New York (1956). S. G. Mikhlin, Integral Equations. Pergamon, London (1957). N. I. Muskhelishvili, Singular Integral equations. Noordhoff, Groningen (1953). R. Walter, Real and Complex Analyses. McGraw-Hill, New York (1974). Y. H. Chen, A method for conformal mapping of a two-connected region onto an annulus. Appl. Math. Me&
4(6), 853-860 (1983).