Optimum design of electrode and insulator shapes

Optimum design of electrode and insulator shapes Hajime Tsuboi

Department of Electrical and Electronic Engineering, Okayama University, Okayama 700, Japan In the optimization of electrode and insulator shapes, any one or more of the surfaces are modified by the iteration methods in order to obtain desired electric field strengths at any electrode and/or insulator surfaces. The object function which is used for modification of the surfaces takes a form of least squares. An object function is evaluated by using integrals over the surfaces on which desired electric field distributions are given, and it is minimized iteratively by the Newton method. The expression of partial derivatives of the object function in the Newton method is formulated using the coefficient matrix of final simultaneous equations of the boundary integral method so that the computation steps can be reduced Key Words:

optimization, electric field, electrode, insulator, Newton method, boundary integral method

INTRODUCTION In the insulation design of high-volatage equipment, optimization of electrode and insulator shapes is performed to reduce the maximum electric field strength or to obtain the desired electric field distribution 1.2,3. The electric field evaluated "in shape optimization is computed by a boundary integral method 4'5. The curved surface elements which provide good approximation of boundary surface are used in the boundary integral method. Therefore, fifth-degree curved triangular elements 6 are used for three-dimensional problems, and cubic-function curved line elements 7 are used for twodimensional and axisymmetric problems. In this report, the optimization method of electrode and insulator shapes using iteration methods is described. Any one or more of the surfaces are modified by the iteration methods in order to obtain desired electric field strengths at any electrode and/or insulator surfaces. The object function which is used for modification of the surfaces takes a form of least squares. Then, an object function is evaluated by using integrals over the surfaces on which the desired electric field distributions are given, and it is minimized iteratively by the Newton method. The expression of partial derivatives of the object function in the Newton method is formulated using the coefficient matrix of final simultaneous equations of the boundary integral method so that the computation steps can be reduced. Some optimization results of three-dimensional, twodimensional and axisymmetric problems are presented for verification of the proposed method.

BOUNDARY INTEGRAL METHOD

(a)

(b)

Electric field calculation is performed by a boundary integral method 4'5 using curved surface elements. The Paper accepted April 1990. Discussion ends November 1990 © Computational Mechanics Publications 1990

curved surface elements which provide good approximation of boundary surface are fifth-degree curved triangular elements 6 for three-dimensional problems and beam elements (cubic-function curved line elements) 7 for two-dimensional and axisymmetric problems. As shown in Fig. 1, the modification of the surface using quadratic elements generates an edge at each end of the element in two-dimensional problems. Therefore, the surface of each curved surface element should be defined so that there is only one normal vector for each point on the boundary between two elements.

Fig. 1. Modification of the contour using twodimensional elements, (a) quadratic element, (b) beam element

Engineering Analysis with Boundary Elements, 1990, Vol. 7, No. 2 83

Optimum design o f electrode and insulator shapes: Hajime Tsuboi In the boundary integral method, the potential Vi and the electric field Ei induced at computation point i are given by

ffoOds

Vi= 1

CO

(1)

S

lSS

= --

C0

o gradq~ ds

(2)

where 0 is the fundamental solution of Laplace's equation, S is the total area of the boundary surface, a is the surface charge density, e0 is the permittivity of free space, r is the vector from the source point to the computation point, and r = [r [. The electric field on the boundary surface is given by

Ei =+nioi _

2eo

lfl

a grade ds

Co s

El=

nioi

Co

(on the insulator surface)

(3)

(on the conductor surface)

(4)

where n; is the unit normal vector at i. The flux continuity condition at i on the insulator surface is given by

(elEli -

Eio~~dv1

di

d i + l ~ ~ ~

e2E2i) ° ni = 0

(5)

where Eli and E2i are electric fields in dielectric 1 and dielectric 2, respectively, el and e2 are the permittivities of dielectric. When applying the potential condition given by equation (1) to the node on the electrode surface and flux continuity condition given by equation (2) to the node on the insulator surface, the final simultaneous equations for the surface charge densities at the nodes on the boundary surfaces are obtained by [C] {o} = [vl

(6)

where [C] is the coefficient matrix having N rows and N columns, {a} is the vector of unknown surface charge densities, and N is the number of unknown surface charge densities. Iv} is given by {v] = {V1V2... Vi... VNc0...0l r

(7)

where Nc is the number of nodes on the electrode surfaces.

Fig. 2. Modification proportion to stress

o f the electrode contour in

where k is the constant, f~ is the stress, e is the permittivity, Ei is the electric field strength and m is the unit normal vector. The stress-ratio method is effective to reduce the maximum electric field strength. But the desired electric field distribution cannot be obtained by the method. Furthermore, it is difficult to fix tl,~ nodes which are located at the end of the area to be modified.

Newton method When the Newton method is applied, an object function is introduced. The object function is evaluated by integrals over the surfaces on which desired electric field distribution are given, and it is minimized by the iteration method. The object function W is given by the form of the method of least squares as W=

L ~ (Ei-Eio) i=l

L 2= ~ w 2= i=1

{wlr{wl

OPTIMIZATION

METHOD

The optimization of electrode and insulator shapes is performed by an iteration method. In order to obtain a distribution of the desired electric field strength, any one or more of the surfaces are modified by using the iteration methods.

wi = Ei - Eio Ei is the computed electric field strength, E~o is the desired electric strength, and L is the number of nodes at which desired electric field strengths are given. The design variables are displacements of the nodes on the electrode and insulator surfaces to be modified. In the method, the direction of the modification is fixed to that of the normal vector in order to reduce the number of the design variables. The design variable vector {xl is given by

di = - k f i = --

k e E2ni

l ° w

(8)

(10)

where xj is the displacement of the node j, and M is the number of the nodes to be modified. Here, we consider a problem to find Ix} which provides the minimum value of the object function. Therefore, Ix} is determined so that the following equation is satisfied.

Stress-ratio method One of simple iteration methods is the stress-ratio method 1,5. The displacement of the electrode surface is determined in proportion to the stress (force) shown in Fig. 2. Therefore, the displacement vector di at node i is given by

(9)

where

[XI = [X1X2... X j . . . XM} T

84

Ei+l

S

ow

ow

T

ax2 "'" axj "'" aXM) =0

(11)

Using the Newton method for multivariable problems in order to solve equation (11), [x] at (k + 1)th step in the iteration method is written as {xl k+l = {x] k _ ([Q] k ) - I {j] k

Engineering Analysis with Boundary Elements, 1990, Vol. 7, No. 2

(12)

Optimum design o f electrode and insulator shapes: Hajime Tsuboi where [J] is the Jacobi matrix given as

where

'ofl

of~-]

I

aXl

[Q]=

F_OI4~I aWl

"'" O X M

(13)

#;...o);,

I

/i~,i

[~ =

0wi

aWll

(19)

o~,L/

L~...~x. Replacing f with VW equations (12) and (13) are rewritten as [X] k+l = {X} k -- ([G] k ) - l v w k

(14)

Therefore, [G] can be approximated by [G] ~ 2[J] r[j]

(20)

And [d} k is calculated by [d} k = (2 [J] X[j])- 1VWk

x~

tol = [

~.@

.....

LaxMaxx

aXxaXz "'" aX~aXMI Ui,¢

....

(15)

o~i;/ I

axMax2"" aXM~ J

Ix} k+l = IX } k + otkld] g

(16)

a k is determined as shown in Fig. 3. In the quasi-Newton method, {d} k is calculated by [d}k= -- [H] k v W g

(17)

where [H] is calculated by the formula of the DFT method s . For the object function which is given by the form of the method of least squares, the Gauss-Newton method can be applied. Then, V W is written as aWj

v w = 2 Z wJ~-~x...2 Z wj

In the practical calculation, [J] is obtained by numerical procedures. From equations (9) and (19), [J] is written as

[s-J = [a [wl alw]

where [G] is the Hessian matrix. The correction of {x} is obtained by applying the formula of the Gauss-Newton method or the quasiNewton method to equation (14). In order to evaluate the Hessian matrix [G], a numerical differentiation is required because the matrix is not obtained analytically. But, in the case that numerical differentiations are used, it is better that only the direction is calculated as {d] k and the correction of Ix} is determined by a linearsearch method because some errors may arise from numerical differentiations in calculation of [G]. The linear-search method using a parameter o~ is given by

M

(21)

aWj] T

= 2[Jl r[w}

a{w}] Lax, ax2 "'" axMJ

_ [a {El a {El

a [El]

(22) Ox2 "'" ~7-£J The electric field [E} to be evaluated is given by using equations (2)-(4) as

L-a~

[E} = [F] [o}

(23)

where [F] is the matrix having N rows and L columns and {a] is the surface charge density vector obtained by the differentiation of equation (6). From equation (23), a[El/OXm is written as O[E} OXm

a[r] O[o} {o-] + [F] - OXm OXm

(24)

On the other hand, the following equation is obtained by the differentiation of equation (6). O Is} = [C] - I O [___~qIs} OXm OXm

(25)

Substituting O{o]/Ox,, for equation (25) in equation (24), O{El/aXm is given by

ore]

(otr]

o[c3~

3Xm = k-~Xm + [F] [C] -1 -ff~--Xm ] {o]

(26)

And alE}fax,, for the calculation of [J] is obtained as (18)

O{EI = ( 0 [ ~ + OCq~ OXm \-~Xm [F] [C] - ' -~--~-x~/ {o]

(27)

where O[E} O[El = [T]. 3 Xm 3 Xm [F] = [T] • [F]

W

[T] =

(29)

0

E21[ E21

0

0

(30)

EL

LI

alE]lax,. and a[C]/aXm are calculated by numerical differentiations as follows: a [F] k+ 1 _ [F(x~ + AX~)] -- [F(x~)] (31)

I I

OXm

0t k

Fig. 3.

(28)

Minimization o f the object function

ax~

a[c] k+l

[C(x~ + aXe)] -- [C(x~)]

OXm

a x*.,,

(32)

Engineering Analysis with Boundary Elements, 1990, Vol. 7, No. 2 g5

Optimum design o f electrode and insulator shapes." Hajime Tsuboi Using equation (27) in order to evaluate [J], the number of times of field calculation can be reduced to one from M. [el in equation (27) is obtained by the field calculation, and [F], [C] and their derivatives which are evaluated by numerical differentiations are obtained in the process of the field calculation. The iteration for the optimization is terminated by the convergence of the object function. The convergence is decided by the normalized value of the object function as follows:

~Wk/L-~Wk-I/L


(33)

Gauss-Newton method (M=4) Gauss-Newton method (M=8) ---o--- quasi-Newton method (M=4) --a-- quasi-Newton method (M=8)

--A-

1 0 "I

.

[..r3

2..,.. "'-..

x

",

"--

(Eio)max

10-2

where (Eio)max is the maximum value of desired electric field strength. The value of e is normally 0.01.

"o.. X

k

10

COMPUTATION RESULTS In order to verify the effectiveness of the proposed method, three models were investigated9: a coaxial cable model, a rod electrode model and a sphere electrode model. In the three models, uniform electric field is given as the desired distribution.

Coaxial cable model The coaxial cable model, which is a two-dimensional insulator model, is shown in Fig. 4 for one fourth part of the coaxial cable. The desired electric field strength on the insulator surface is equal to 12.01 (V/unit length) which is the theoretical value. The initial and optimal shape of the insulator are the ellipse and circle shown in Fig. 4. In this model, contour A - B without node A is modified and electric field strengths at the nodes on contour A - B are evaluated. Figure 5 shows the changes in the normalized values of object function. It was verified from Fig. 5 that the Gauss-Newton method gives very fast convergence independent of the number of design variables M, and the quasi-Newton method gives steady convergence whose speed was in proportion to the number of design variables. Table 1 shows the computation results of the coaxial cable model.

I i

]

optimal shape

e.s 1 [ '

~

/

initial shape

-I "* .~'.~

iii!i!:!:!:!:!ii::. """:::::i::::: ::-:':-::i::i::

IFig. 4. 86

Coaxial cable model

""A

o

\

-3

\

0

,

X

'

,

1

2 3 4 iteration step

,

,

5

6

Fig. 5. Changes in the value o f the object function for the coaxial model Rod electrode model in cylinder The rod electrode model in a cylinder with a closed end, which is a axisymmetric model, is shown in Fig. 6. The desired electric field strength is given on contour A - B and is equal to the theoretical electric field strength on inner conductor of a coaxial cylinder model with radii of 5 and 7 unit length. Contour A - B without node A is modified in model 1 and contour C - D without node C is modified in model 2 so that the electric field strength on contour A - B become equal to the desired electric field strength. In Fig. 6, solid lines indicate optimal contours. The optimal contour of the model 1 was obtained by the Gauss-Newton method, and that of the model 2 was obtained by the quasi-Newton method. The iteration for the model 2, in which the evaluation area of electric field strength is set apart from the modification area of contour, did not converge by the Gauss-Newton method because the values of the elements of the Hessian matrix were small and could not be obtained with accuracy. Figure 7 shows the electric field distribution on the contour A - B . As a result, uniform distributions of the desired electric field strength were obtained as shown in Fig. 7. Table 2 shows the computation results of the rod electrode model. Table 1.

¢',11

"~,.. """

Computation results o f the coaxial cable model Max. error

of Ei

(%)

CPU time (sec)*

Method

N

L

M

Iteration steps

Gauss-Newton method

15 27

5 9

4 8

2 2

0.638 0.129

1.2 3.0

quasi-Newton method

15 27

5 9

4 8

4 6

0.744 1.68

2.3 7.8

N: L: M: *"

the number of unknowns the number of nodes at which electric field is evaluated the number of nodes of which position is modified using NEC ACOS-2010 (47 MIPS)

Engineering Analysis with Boundary Elements, 1990, Vol. 7, No. 2

O p t i m u m design o f electrode and insulator shapes: Hajime Tsuboi

....

Z

initial shape optimal shape

-

100v

I

I

lOOV

_l[

i 7

i5 I

A

I

I Iq I

,,_1

2 (a) Y

D r"---

I

I (a)

0v

(b) I

Fig. 6. R o d electrode model in cylinder, (a) m o d e ( l , (b) model 2

Table 2. Computation results of the rod electrode model in the closed cylinder Model

Method

N L M

Iteration steps

Uniformity**

1

GaussNewton

41 15 14

3

1.002

6.9

2

quasi Newton

36 14 14

7***

1.014

14.0

**"

-~x

i

(b)

Fig. 8. Sphere electrode model in cube, (a) initial shape, (b) cross section at z = 0

(Ei)max/ ( E i ) m i .

**k: t h e v a l u e o f e is 0 . 0 0 1

0-0-0-0-0-0-0-0

70

--o-- initial shape optimal shape

70

I

CPU time (sec)

I I I I I

!

9

O a

/ 60

! 2..

\% k..-tt

I # .t, 0I

/ --

60

'

,

50

A

B

. . . . .

t t

I I I I

t I%

o

-

I I

/ ,o

b" 50

contour (a)

Fig. 7.

'

A

.....

contour

'

B

(b)

Electric field distributions on the electrode surfaces, (a) model 1, (b) model 2 Engineering Analysis with Boundary Elements, 1990, Vol. 7, No. 2

87

Optimum design of electrode and insulator shapes: Hajime Tsuboi Table 3. cube

Fig. 9.

---o-- initial shape optimal shape

Method

N

L

M

Iteration steps

Uniformity

CPU time (sec)

GaussNewton

192

45

42

3

1.021

924.0

Sphere electrode model in cube The sphere electrode model in a cube, which is a three-dimensional model, is shown in Fig. 8. The desired electric field strength is given on the sphere surface and is equal to 40 ( = 100/2.5) (V/unit length). In one eighth part of the sphere electrode model, the triangular area A - B - C without nodes A, B and C is modified so that the electric field strength on the triangular area A - B - C becomes equal to the desired electric field strength. Figure 9 shows optimal shape of the sphere electrode obtained after three iteration steps. Figure 10 shows the electric field distributions for the initial shape and the optimal shape along contour A - B and contour A - D . It became clear from the computation of the sphere electrode model that the Gauss-Newton method provides good convergence for the three-dimensional problem the same as two-dimensional and axisymmetric problems. Table 3 shows the computation results of sphere electrode model.

Optimal shape o f the sphere electrode

50

Computation results o/" the sphere electrode model in the

#

40

30 CONCLUSION

20

!

!

A

contour

B

(a) 50

40

N

30

20

' A

contour

D

(b) Fig. 10. Electric field distributions on the sphere electrode surfaces, (a) contour A - B , (b) contour A - D 88

The method for the optimization of electrode and insulator shapes in two-dimensional, axisymmetric and three-dimensional problems was presented. The conclusions can be summarized as follows: 1. In order to obtain the shape which provides the desired electric field strength, the object function which takes the form of least squares was introduced, and it was minimized by the Gauss-Newton method or quasi-Newton method. 2. The expression of partial derivatives of the object function in the Newton method can be obtained by using the coefficient matrix of final simultaneous equations of the boundary integral method and it is evaluated by the numerical differentiation. 3. The Gauss-Newton method gave very fast convergence independent of the number of design variables, but did not converge when the evaluation area of electric field strength was set apart from the modification area of contour. 4. The quasi-Newton method gave steady convergence whose speed was in proportion to the number of design variables. The desired electric field distribution was obtained by the iteration methods and the maximum field strength was consequently reduced. 5. Because any surface can be chosen as the surface to be modified or to be evaluated, the method is practicable for the shape optimization of electric machinery and equipment.

Engineering Analysis with Boundary Elements, 1990, Iiol. 7, No. 2

O p t i m u m design o f electrode a n d insulator shapes: H a j i m e T s u b o i computer in three-dimensional potential problems, Boundary Elements X, Vol. 1 (Ed.: Brebbia, C. A.), 1988, 363-377 5

REFERENCES 1

2

3

Misaki, T., Tsuboi, H., Itaka, K. and Hara, T. Optimization of three-dimensional electrode contour based on surface charge method and its application to insulation design, IEEE Transactions on Power Apparatus and Systems, 1983, PAS-102, 1687-1692 Tsuboi, H. and Misaki, T. The optimum design of electrode and insulator contours by nonlinear programming using the surface charge simulation method, IEEE Transactions on Magnetics, 1988, 24, 35-38 Tsuboi, H. and Misaki, T. Optimization of electrode and insulator contours by using Newton method (in Japanese), The

Transactions of the Institute of Electrical Engineers of Japan, 4

1986, 106-A, 307-314 Tsuboi, H. and Ishii, Y. Numerical integration for super-

6 7

8 9

Misaki, T., Tsuboi, H., ltaka, K. and Hara, T. Computation of three-dimensional electric field problems by surface charge method and its application to optimum insulator design, IEEE Transactions on Power Apparatus and Systems, 1982, PAS101, 627-634 Yang, T. Y. Finite Element Structural Analysis, Prentice-Hall, London, 1986 Tsuboi, H. and Misaki, T. Boundary integral method for twodimensional and axisymmetric electric field problems, Theory and Applications of Boundary Element Methods, (Eds: Tanaka, M. and Du, Q.) 1987, 207-216, Pergamon Press Gallagher, R. H. and Zienkiewicz, O. C. (Eds) Optimum Structural Design, John Wiley and Sons, London, 1973 Tsuboi, H. Shape optimization of electrode and insulator, Boundary Elements X, Vol. 1 (Ed.: Brebbia, C. A.), 1988, Springer-Verlag, 603-617

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