The optimization of plane-parallel magnetic fields

0041-5553/77/0601-0217$07.50/O

U.S.S.R. Comput. MathsMath. Phys. Vol. 17, pp. 217-222 0 Pergamon Press Ltd. 1978. Printed in Great Britain.

THE OPTIMIZATION OF PLANE-PARALLEL MAGNETIC FIELDS* 0. I. LIETUVIETIS,

G. A. RADZIN’SH and U. E. RAITUM Riga

(Received 7 July 1975) THE PROBLEM of the optimization

of the perturbation

in a direct current magnetic field is formulated

in a set of plane curves. A method of approximating problem, for which gradient minimization calculations

introduced

by a ferromagnetic

as the problem of the minimization

body

of a functional

the original problem by a finite-dimensional

methods may be used, is proposed, and the results of

are given for one simulated problem.

1. We consider the following problem of the optimization of a plane-parallel tield by the choice of the shape of the ferromagnetic body for a given distribution state currents and for a linear dependence H in the ferromagnetic

B = nH of the magnetic induction

magnetic of the steady-

B on the intensity

material.

In the Oxy plane we are given a section S by the plane z = 0 of conductors of currents 6 (x, y) flowing in the direction of the Oz axis. The set S is bounded, value of the function 6 (x, y) over S is zero. The functional

ZAZ(B)=

JJFb,

Y,WZ(~,

Y),-W,(T

y))dx

and a density and the mean

dy,

D

is given, where B= (B,, B,, 0), the bounded set S, the function F is continuously dependent differential to determine

domain D is situated at a positive distance from the

differentiable

on B, and is piecewise-continuous

with respect to its arguments,

is explicitly

with respect to (x, y), and L, and L2 are given

operators with smooth coefficients,

in particular,L,B,=B,,

LzB,=B,.

It is required

the contour E, enclosing a section !Ylof the plane z = 0 of the ferromagnetic

such that the induction

B of the perturbed

magnetic field minimizes the functional

body,

Z(B).

In the general case the problem may not have a solution. Hence in what follows we will consider only the question of the construction which the functional

I decreases monotonically

of a sequence of contours (the induction

field for specified current densities 6 (x, y) and the magnetic permeability is uniquely

determined

ik, k=O, 1,. . . , on

B of the perturbed

magnetic

u of the ferromagnetic

by the contour I). Here and below it is assumed that B = H outside

the set a. 2. We f= the interval [a, b] and for an arbitrary a>0 we denote by V(a) the Banach space of pairs of functions (s(t), y(t)), t=[a, b], which belong to the space Cl,,[a, b] and (x(a), y(a))=(~(b), y(b)), (i(a), ~(a))=(ri:(b), ~(b)).IfthecontourZisacurveofthe Lyapunov class C, ,o, then it has a parametric representation by means of the function

*Zh. vjbhisl. Mat. mat. Fiz., 17, 3, 780-785,

1977.

217

(2 (t) , y (t) )

0. I. Lietuvietis, G. A. Radzin’sh and U.E. Raitum

218

E V(a). In this case the induction of the perturbed magnetic field has the representation (see, for example, [ 1] )

(1)

h

A(x,Y)=~~o(t)ln~df+~~~S(x’,y’)ln~dx’dy’, r

T

a

D

where r is the distance between the point (x, y) and the point of integration, and the function o(t) satisfies the integral equation (,

h +n

ss

(2)

H(x (11,YW, x’, ~'16 WY') dx’ a~',

D

K(x(t)

I

Y(t) g ll)_ ,,

[F;-s(t)l5i(t)-[9-y(t)ll(t) [E-s(G12+[11-Y(t)12

’

x= -==I p-l P+i

1.

Since Eq. (2) is a variant of‘the equation for the normal derivative of the potential of a simple layer and is uniquely solvable for o(t) =C[a, b] for any free term of C [a, b] , then the choice of a pair of functions (s(t), y(t)) ET’(U) by means of the representation (1) uniquely determines the value of the functional I, that is, I--I( (x(t), y(t))), provided that this pair of functions deilnes a Lyapunov curve not having points in common with the sets S and D. 3. Let some Lyapunov curve 1, with the parametric representation JZO(t), YO (t)) E V(a). have been chosen. We specify the function 9 (r) =C[u, b] as the solution of the integral equation

9(t)--

h n

b

s

~(xo(s),Yo(s),xo(~),Yo(~))~(S)~s

(3)

-

F,‘,,,(x, y, -ML,

LB,) Lz ;

In

L r

dx dY, 1

where B, and By are determined by the chosen (x0 (t), y. (t)), r is the distance between the points (x0(t), y. (0) and (x, Y), and the differential operators L, and L, act on the variables (x, y). Here the free term of Eq. (3) is the FrechCt partial derivative of the functional I with respect to a(t), on which it depends by way of the representation (1).

Let the functions (F~(t), ye(t)) E V(a) define a Lyapunov curve and let the domain bounded by it be situated at a positive distance from the sets D and S. Then at the element (x0 (t), y. (t)) the functionalI((x(t),y(t)) has a first Frechdt derivative with respect to (s(t), y(t)) =V(a) and this derivative is identical and with the first Frechet derivative of the functional

Short communications

(x, Y, b;

I(x(t),y(t))=JJF

+

r 1, -L,; ; Jaao(t)hkdt I) b

1

8

2n JJ

1

+-

[~jlll+&4f)dl a

DO +-

2n JJ 6

6 (x’, y’)ln __!_ dx’ dy’ r

[

6(x’, y’)lnLdx’dy’

dx dy

r

8

JJ

with respect to (r(t)),

219

K(x(t), y(t), x’, y/)6(x’, y’)dx’dy’

y(t)) =V(a)

1

dt

at the element (~~(f),_~u(t)).

Here so(t) is the solution of

Eq. (2) with (x,,(t), y. (t)), the operators L, and L2 act on the variables (x, y), and r is the distance between the points (x, y) and (x’, JJ’) or the points (x, y) and (x(t), r(t)) ProoJ The properties of Lyapunov curves and the definition

respectively.

of the space V(a) imply that

the operator b 9(x(t),

depending

on (x(t),y(t))

y(t))a(t)=

J a

K(x(t),

(t),defined

is FrechCt differentiable

yo (t)

with respect to (z(t),

y(t) ) =V(a)

(see, for example, [2] ), the implicit function

by Eq. (2). m some neighborhood on (20 (t),

al-, eta, bl,

y(s))o(4ds:C[a,

as p ammeter, is FrechCt differentiable

on the element (x0(t), y. (t) ), consequently a(x(t),y(t))

Y (t), t(s),

of the element

). Hence, from the properties

(x0(t), YO(t) ) E V(a) ,.

of conjugate integral

operators and the fact that the contour lo is at a positive distance from the sets S and D, the statement

of the lemma follows.

Remark.

An analog of Lemma 1 also holds for some contours

and for functionals

I defined

by an integral along a fixed curve. Lemma 1 implies that it is possible to use gradient methods to minimize the functional 1(x(t), y(t)).Since the FrechCt derivative of the functional 1(x(t), y(t))as an element of the space conjugate to V(a), is in the general case not a summable function but can be calculated for any futed direction (x(t) -50 (t) , y(t) -I/O(~) ) , Therefore in the practical realization of the process of minimizing the functional Z it is advisable to confine ourselves to a finite-dimensional linear space of functions of (t(t), y(t) ), possessing a basis of elements in whose direction the derivative of the functional

I is comparatively

well computable.

In essence this requirement

involves two features: the basis functions must be fairly simple and smooth, and each of them must be equal to zero outside some small interval of [a, b] , since the derivative of the functional Z in the direction Ax(t) (or Ay (t)) contains an integral over the set where the function Ax(t) is non-zero. The latter property of the basis functions is also due to the need to vary the contour 1 in the neighborhood of an arbitrary point without changing it far from this point.

i=l,

4. We fix the segment [a, b] , the numberN, the points a+h=ti< . . .
2,.

ti+i_ti=2h

220

0. I. Lietuvietis, G. A. Radin’sh and U. E. Raitum

I(

c2(2h-~-t~l)3+c2(26-171)~+c3(2h-~-trl)

‘pi(t) =

h 4 1t-ti

4h-It-t*1

-C5

h

3hGIt-ttI<4h,

’

)

4h
cz=c5=~/zo,

cs=ck=*/&.

c I , . . . , c5 the basis functions

With this choice of the coefficients with their first derivatives and qi(tj) e&j, The parametric 2Nnumbers

1G 3h,

2

0, Cl=g/zo,

(

representation

(z(t),

where 6ii is the Kronecker y(t)

are continuous

together

delta.

) of a specific curve I is defined by means of

(G, . . . , zN), (y,, . . . , yN) : N

x(t) =

c

xicpi (t) + Xicpl(t-2Nh) + r2fp2(t-2Nh)

i=l

+ IN--irpN--i(t+2A%)

+ zacps(t+2Nh),

(4) N

c

Yicpf

Y(t)=

0) + Yicpi(t-2Nh) + y292(t-2Nh)

i-i

-I-

The functions

(x(t),

YN--I(PN--I(t+zNh) -I- YNCPN (t+zNh).

y(t)) defined by formulas (4) belong to the space V(o), they specify a

yN) of the Oxy plane, and Zis a closed curve I passing through the points (z,, JJ~),. . . , (IN, Lyapunov curve provided that the “crude” specification of the contour 1 by the points (xi, yi)

i=l,

2 , . . . , N,is sufficiently

good.

Therefore, the problem of minimizing

the functional

Z((x (t), y(t)) ) in the set of plane

curves by means of the representation (4) is replaced by the problem of minimizing the functional Z=Z (X1,. . . , zN, yi, . . . , yN) in a Euclidean space EzN of dimension 2N. It follows from the lemma that if the curve (IO(~), ye(t) ) , defined by formula (4) with the parameters (x10,. . . , XN.O, yia, . . . , yNO), is a Lyapunov curve, then the partial derivatives of the functional Z(xr,.. . , XN, yi,.. . , yN), calculated at the point (xiO,. . . , XNO, ~~0,. . . , yNO),are d ZXi’ = r(xo(t) + zqa(t), YOW) I r-0, dz

i=3,4,.

. . , N-2,

d ZIli’ = J(x0 (t), Yo 0) + WJf (0 1 dT

i=3,4,.

. . , N-2.

The partial derivatives for i=l,

2, N-l,

I r-o,

N are calculated

similarly, in accordance with the

nature of formulas (4). The expressions obtained for the partial derivatives permit gradient methods to be used to minimize the functional Z(xl, . . . , zN, ~1,. . . , yN), it being necessary at each step to solve the integral equations (2) and (3) and evaluate the multiple integrals determining the derivatives ZXi’, Zbri’.If the set S consists of a finite number of domains of simple

Short communications

221

form and the function 6 (x, y) is constant in these domains, then the integrals over the set S in the expressions for J2; and 1,; can often be found in explicit form and finally to determine the derivatives Izi’ and I,; it is necessary to evaluate the integrals over D and over la, &I X [k-2k.,

ti+23]. I=

J

(+,)Zdi

1’

t

Y

0 0=-f

0 6=-f

FIG. 1 5, To check the efficiency of the proposed algorithm the problem of minimizing the functional

for a magnetic system, whose initial state is shown in Fig. 1 was considered. Here the crosssection of the ferromagnetic body Q consists of the two components at, and !+, and the set S of four components. The boundary of each of the domains !L?Iand Q, is defined by means of 48 points, symmetry about the coordinate axes being preserved. The calculations were performed for u = 1000. To find the approximate values ui of the function o(t) at the points tip i=l, 2,. . . , N, we replace Eq. (2) by the system of linear algebraic equations

(Ti =

+

UijOj

+

fir

i=l,

2,. . ., N,

(5)

by the method proposed in f3f. The corresponding system of equations for the approximate values pi of the function J/(t) at the points ti, i=l, 2,. . . , N,is as follows:

$i

=

Jl.

ajil)j

+

giy

i=l,2,...,N,

(6)

222

Yu. D. Shmyglevskii

where fi and gi are the approximate respectively.

Equations

values at the point ti of the free terms of Eqs. (2) and (3)

(5) and (6) are solved by Gauss’s method. The approximate

values of the

u(t) and J/(t) in the whole interval [a, b] are found by means of a five-point parabolic

functions

interpolation,

and all the integrals encountered

along the curves I and I’ in the determination

the derivatives I,,‘, I,,’ were evaluated by a four-point

or five-point Gaussian formula.

For the case where the values of the gradient of the functional the boundary

of each of the domains 52, and Q2,8 iterations

of the functional

I on the curve I, was 1=0.103.10-‘,

of

I were taken at 20 points on

were performed.

The initial value

and on the curve Z8 the value was 1=O.2O2.1O-3

(the contours lo and I, are shown in Fig. 1). The values of the function By (x,y) on the segment I’ were found in the strip -2.739+,<-2.052

for the contour Z. and in the strip -2.152<&,<-2.146

for the contour I,. Translated by J. Berry REFERENCES

1.

TOZONI, 0. V. Calculation of electromagnetic fields on computers (Raschet electromagnetitnykh vychislitel’nykh mashinakh), “Tekhnika”, Kiev, 1967.

2.

KANTOROVICH, L. V. and AKILOV, G. P. Functional analysis in normed spaces (Funktsional’nyi analiz v normirovannykh prostranstvakh), Fizmatgiz, Moscow, 1959.

3.

DZERGACH, A. I. and RADZIN’SH, G. A. Calculation of the two-dimensional field of electromagnets with unsaturated iron by means of integral equations. Tr. Radiotekhn. in-ta Akad. Nauk SSSR,

polei na

No. 14,70-75,1973. U.S.S.R. Comput. MathsMath. Phys. Vol. 17, pp. 222-228 0 Pergamon Press Ltd. 1978. Printed in Great Britain.

A VERSION

OF THE MOMENT

THE TRANSFER

0041-5553/77/0601-0222$07.50/O

METHOD

OF SELECTIVE

OF CALCULATING RADIATION*

Yu. D. SHMYGLEVSKII Moscow (Received 18 March 1976) A MOMENT method using Laguerre frequency transfer for arbitrary optical cell dimensions calculation numerical

polynomials

is used to calculate the radiative

and large temperature

drops. The results of the

of radiative transfer in an isothermal half-space illustrate the accuracy of the scheme.

In [l]

the moment method is used for the transfer equation in a differential

form and

enables calculations on cells with small optimal dimensions to be performed at all frequencies. An improvement of the method was described in [2] . The results of [3,4] permitted the method to be extended [5] to the case of arbitrary optical dimensions of the cells. The version of [S] permits calculations to be performed for comparatively small temperature differences at adjacent computing points. This is due, for example, to the fact that when radiation from an absolutely black body at the temperature T2 falls on a particle at a temperature T, , it is necessary within a semiinfinite interval of measurement of the frequency v to expand in special polynomials the ratio of Planck functions B (Y, T,) /B (Y, Ti), which for T+T, increases exponentially with frequency. The stability of the method is proved in [6] , and the method itself was used to calculate the flow of air in a circular tube with transparent walls [7]. The method is further improved here. *Zh. vychisl. Mat. mat. Fir., 17, 3, 785-790,

1977.