00415553/77/06010217$07.50/O
U.S.S.R. Comput. MathsMath. Phys. Vol. 17, pp. 217222 0 Pergamon Press Ltd. 1978. Printed in Great Britain.
THE OPTIMIZATION OF PLANEPARALLEL 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 finitedimensional
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 planeparallel 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 piecewisecontinuous
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, 780785,
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)[9y(t)ll(t) [Es(G12+[11Y(t)12
’
x= ==I pl 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, II( (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 finitedimensional 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 nonzero. 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(26171)~+c3(2h~trl)
‘pi(t) =
h 4 1tti
4hItt*1
C5
h
3hGItttI<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(t2Nh) + r2fp2(t2Nh)
i=l
+ INirpNi(t+2A%)
+ zacps(t+2Nh),
(4) N
c
Yicpf
Y(t)=
0) + Yicpi(t2Nh) + y292(t2Nh)
ii
I
The functions
(x(t),
YNI(PNI(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 r0, dz
i=3,4,.
. . , N2,
d ZIli’ = J(x0 (t), Yo 0) + WJf (0 1 dT
i=3,4,.
. . , N2.
The partial derivatives for i=l,
2, Nl,
I ro,
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 [k2k.,
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 fivepoint 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 fourpoint
or fivepoint 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.2O2.1O3
(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 twodimensional field of electromagnets with unsaturated iron by means of integral equations. Tr. Radiotekhn. inta Akad. Nauk SSSR,
polei na
No. 14,7075,1973. U.S.S.R. Comput. MathsMath. Phys. Vol. 17, pp. 222228 0 Pergamon Press Ltd. 1978. Printed in Great Britain.
A VERSION
OF THE MOMENT
THE TRANSFER
00415553/77/06010222$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 halfspace 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, 785790,
1977.