U.S.S.R. Comput.Haths.Math.Phys.,Vol.23,No.6.pp. 75-79,1983 OO41-5553/83 $iO.OO+O.00 Printed in Great Britain �9 1985 Pergamon Press Ltd.

A DOUBLY PERIODIC PP, OBLE~,I OF ~IAGNETOSTATICS*

S.T. TOLMACHEV

Integral equations are obtained for the magnetic field in a regular spatially two-dimensional heterogeneous medium. A method for the numerical solution of the basic equation is proved, and is particular, the bounds are obtained for the method of simple iteration to be applicable. The rate of convergence of the iterative process is expressed in terms of the physical properties of the medium, which in general is assumed to be non-linear, inhomogeneous, and anisotropic.

A medium with regular spatially two-dimensional structure, formed by a set of congruent groups of magnetic materials (Fig.l), will be described in the complex E plane. To every group of bodies there corresponds a bounded (in general multiply connected) domain S~n~USm$

with sufficiently smooth boundary r..~ur=d,]=t, 2 .... , k, m, n=0, ~|,.... The domain S00, cor- responding to the main parallelogram Q with periods ~, and oz (henceforth we assume that Im~t= 0, Imx>0, x=~z]o,), is denoted for convenience by S. Accordingly, F=F00, S=S+F. The domain outside the magnetic materials is denoted by S,~E\US~,.

Denote by B, H, and J respectively the magnetic induction, field-strength, and magnetiza- tion. The inhomogeneous, non-linear, and anisotropic properties of the set of congruent elements j is specified by the function B~=BJ(H,z),z~US=,. If z~S,, then B=[toH.

Assuming that we know the density 6(z), im6~0, of the currents distributed inS, according to a doubly periodic law, we find the regular functions B(z) and H(z), satisfying inEMaxwell's equations

OB.__..+ OB..__.y= 0 OHv 0 1 t , = 6 ax Oy ax oy ' (1)

and on the boundary F the mating conditions

B,.=B.., H.=H.. (2)

The subscripts i and e refer to the limiting values from inside and outside of the normal and tangential components of the vectors B and H on the boundary F.

This problem has many practical applications. It covers the reduction problem for a heterogeneous medium, which has a vast literature, see e.g., /I/, the problem of electro- magnetic sounding and diffraction by periodic arrays, and the study of the local and integral properties of composite materials, etc.

The approach to the solution of the problem given below is based on the application of integral equations and functions of a complex variable, The method of integral equations was developed in /2, 3/. Analysis of field problems by methods of the thoery of analytic func- tions was described in /4, 5/, etc. A distinctive feature of the present paper is that account is taken of the doubly periodic structure of the field, and of effects due to the non-linear properties of the medium. The problem is reduced to a non-linear singular integral equation (s.i.e) with respect to the domain S, while the method is justified by using the theory of generalized analytic functions (g.a.f.) in Vekua's sense /6/.

, o / t a: I [

, / 0 ) 2 / /

! /

i. We first prove some auxiliary rela- tions. Let dipoles with the same moment M be located at points %~m0d(ol, 02). The complex potential is

(z) = M___ ~ (z-U +Cz, ( l . 1) IV

and the field-strength is

Fig. 1

C=] ImM F

- - .~ , _ ~ U - - _

l l ( z ) = - W ' (z)=-5-E= ~ (z-U-c=~-~'(z-U-c, (1.2)

where ~(z--~), ~(z--~) are Weierstrass func- tions, and C is a constant, easily found from the conditions for double periodicity of the potential ~=Re|V(z):

M ~1, 1 2~ o~ 2,~ MQ. (1.3)

*Zh.vgchisl.Mat.mat.Fiz.,23,6,1402-1409,1983 75

76

In the last relation, F is the area of the basic parallelegram of periods the diagonal matrix

0=II ~ 0 ~, and Q is

Starting from relations (i.i) and (1.2) for|V(z) and H(z), we introduce the integral oper- ators

8

where P=JJ(~)drt is the complete dipole moment of the domain S. B

With or, oz =~, the above operators TJ and l]~J reduce to the operators of /6/, namely:

T. i C J(~) . i e J(~) d j= - - |-----q- ~, -- __ _ _

Notice that the integral in (l.4b) is singular. In /6/ the relations O, TJ=HJ, @~T.I=J are proved, which can be extended to the doubly

periodic case. For, since the function ~(u)contains a simple pole in the basic parallelogram , the function Y/(I/(u--o)-}-I/o+u/o 2) is analytic in it, and hence the function

o,z'/ t + • i 1 .1 ~u-0~ o o ' ! " (u -o ) ~ ~ J "

i's likewise analytic. Hence

t v , t o,r~l=o,r/+o,-~-~, (. . .)dn--~-PO=IIj. (1.5)

In the same way we find that

O, TtJ---~O, TJ'=i, O~TJ=I, O~TQi=HJ. (1.6)

We can easily prove the identity

s s{f P= /(~)d%= [~l(g)--~ dn, tt 8

in the light of which we can write the function H~I in the equivalent form

r s

Using Green's formula

! O,mT+ i---f J (z)~=O 2q (1.o)

and recalling (1.6) and (1.7), we obtain

O, IIJ=a~fa(O,J)=O,A a, HJ=0ri. (i .9)

2. we shall show that the initial problem can be reduced to an equivalent s.i.e. First consider the linear case of an inhomogeneous isotropic medium. For z~ , we take B(z)= ~to~(z)=~to~t(z)H(z). Then, for z~S, , we have /~(z)=J/(z).

Let z(s)be the parametric equation of the contour F, where z is the affix of the point of curve F corresponding to arc length s. Rewriting (i) and (2) in the equivalent form

211,(t)=(tt+t)ll,(t)--(rt--l)ii,(t)[z'(t)]', t~V, (2.1a)

O, pH+Orp21=O, O,H-O~I]=]6 (2.1b)

and noting the obvious local relation

l(z) =B(z) --U(z) =X(z) [B(z) +H(z)], (2.2)

where ~(z)=[~t(z)--l] [~(z)+l]-t, we obtain

O, (l-'l) +O;i=j6=]O,T=6. (2 . 3)

Eliminating 0~ from this relation using (1.9), we obtain

a, (~,- q+-fl-?l- j ~6 ) =o,

77

or w 1=~(1~,~6-IIJ) +4], (2.4)

where ~ is a harmonic vector. We shall show that s.i.e. (2.4) with ~-0 is equivalent to the basic problem (i), (2).

From the boundary properties of the Cauchy integral, we obtain the equation for the jump of function [I~1 in passing through F (see /6/):

~nj=(nj),-(nJ),=1(z) [z'(s)]', z~r.

Noting that J(z)-----0 and l(z)-----0 in So, and that the function Tal is continuous on F (no surface currents), we obtain

A (~-il) +AIIJ--------A (B+H) +Alibi= (B+H),-2H.--I (z) (z')*= (D+i) H,- (~t-l) H, (z')'-2H,=A~. (2.5)

Comparing (2.1) and (2.5), we see that A~=0, i.e., �9 is continuous on F. But ~(~)=0, and hence, by Liouville's theorem, ~------0. Thus, if l(z) is a solution of s.i.e. (2.6), the functions ~(z) and //(z), connected with l(z) by (2.2) , satisfy (I) , (2) . Hence problem (i) , (2) is equivalent to s.i.e. (2.4) with respect to the magnetization vector J.

Put

where H,(z) is the field strength of the external sources. Thus the intial problem reduces to s.i.e.

:(z) =Z~(~)H,(z) + - ~ [.~(~)~(z-~)+~U(~) ]e~,= g(z)-~.(z) noi, (2.6)

or in operator form: S l (z) =., (z) +~.(z) no~=g(z).

With ~.'=const#l, the volume density of vorticity sources (9,1=0, i.e. vector J is harmonic. By (1.7), for this case

Let R(z, ~)=l(~)~(z--~) and z~St. Since the function ~(z--~) is analytic in S, and hence

@J/=(dJ)~(z-~)=0 for M=e0nst, we obtain from Green's formula

'! :(~) = ~17~ :(~)~(~-~)e]. (2.7)

This last expression is the doubly periodic generalization of Cauchy's formula. Let z~F in (2.5) and (2.6). Using the limit properties of the Cauchy integral and

assuming that ~(z)is continuous in S, we find

I _ , , I i J(z)=g(z)'t-~(z) [-~-Jz (s) +-~-~n~ J(~)~(z-~)d~'4"2-~ Q~r'~J(~)d~]' (2.8a)

l(z)=~i- J(~)~(z--~)d~. (2.8b)

Putting

11(~) d~= [a (~) +/t (~) Iris, (2.9)

where 0 and T are the surface densities of the coupled charges and currents, and noting that, with @,I=0 , the function ~=J(~)~ is analytic, i.e., 0i~=0, we obtain

.[ d:(~)e~ .[[o(~)+j~(~)l~e~=o. (2.1o) P

Using (2.7)--(2.10) and (1.3), we obtain

o(z)=2~Rel,F_'[211,(z)+-~-So(~)(~(z--~)---~t(z-~)+ ~ 1 ds]}, (2. lla)

x(z)=7~Re{li"[--2lll,(z)--~;~(~)(~(z--~)-- ~(z--~)-- 2~(~)) ds]}. (2.11b)

P

Notice that, with or, Oz =~ , (2.11) reduces to Fredholm equations of the 2nd kind with respect to surface densities o and ~.

3. We will now study the integral equations. We consider the basic s.i.e. (1.9). We shall estimate the norm of the function HQ] in space L2(S). By definition, and using rela- tions (1.6),

78

IlllJIl'= ([IJ, rlJ) =~ O,(T-~TJlrIJ) dr,- ~ "T j jo , l l j an . s

By (1.6) and (1.9) , @.~J=], @~nJ=b,J, and hence

Applying Green's formula (1.8) to (3.i) and putting

s s z ] ~

(3.1)

we obtain (nJ, nj) = (i, i) +h.

Let us show that I0~0. In fact,

and hence I I I IJ I l< l lJ l t . The equality is obtained with I0=0. The necessary and sufficient condition for this is

that lq~]~-0 in S,, which, by definition of IInJ, is equivalent to the identities H$~0 for J=~, or B;=--0 for J=--H3 in S. An infinite set of functions ](z)~O exists satisfying

these conditions, and these functions are the eigenfunctions of s.i.e. (2.6), corresponding to the infinitely multiple eigenvalues ~=I and }.=--I respectively.

Thus the norm of the operator 1In in the space L2(S) is unity. From this, we have: for 0<D(=) <:~ }.~<I , the s.i.e. (2.6) is uniquely solvable with any function llo(z), the solution J(z) being obtainable by the method of successive approximations:

A(z )=g( z ) -~ . ( z ) I I~ : . - , ( z ) , n=l,2 . . . . . ( 3 . 2 )

The iterative process (3.2) amounts to.a geometric progression with ratio q=maxl, and its rate of convergence is given by II].-JII<q"II]o-]Jl.

our results can be extended to the case of a non-linear anisotropic medium. It is then convenient to introduce the new variable U=B+H=2H~ As a result, we arrive at the non-linear s.i.e.

U ( z ) = 2 H o ( z ) - - n J [U(z) ] , (3 .3)

where J(U) is the characteristic of the magnetic state of the medium. We introduce the tensor

d: d(2~--~) Ld~ ~ =d(B+U) (l~d--l) (~d+i)-"

where I t d - - - - d ~ l d H is the relative differential permeability tensor of the medium. Since, for physically realizable media, the tensor lid is symmetric and positive definite, the diagonal

< ldJl<ldU[ for any U. tensor I d satisfies the inequality ~li |. Hence

We now easily obtain the inequality

IlnoJ ( u , ) - n ~ : (u,) i I<.ql lU,-U, ii,

\ % -

v . . r ~. ~ . , . / / I I / / / t</.~. . . 4

/

Fig. 2

where q = m a x I~.[<l. Hence the non- linear s.i.e. (3.3) is also uniquely solvable by the method of simple itera- tion, the convergence of the successive approximations being no worse than for the corresponding linear equation, which is majorant for (3.3). Notice also that the convergence of the iterative process is specially rapid when the medium is saturated and ~d~0.

4. We will consider some features of the numerical realization of the method. In practical computations it is best to use the following expansions /7/:

~, , ~l, ~] I h+h- ' , 2 ~-1 qZ'h-" q2"hZ I_r )

~, n ~ t ~ q2"h-' ( I--qZ'h:V ]

(4.1a)

(4.n))

where q=exp (]aT), h=exp (jay), v=u / 6~,= (z-~) / r

To ensure that the domain S can be discretized, we introduce a representation for regular in S. In accordance with (1.5) and (1.6), for ]=c0nst we can write

II~J=lCr+21 Im (Fs I F), where

s 21r is the area of the domain S, and

79

I l a l

We can now write

2j ] 5 8

The integral over S in (4.2) is absolutely convergent. The evaluation of the one-dimen- sional integral Dr (up to boundary value) presents no great difficulties. Its limiting value from inside is

t..'<.,> ]'+ [

It is sometimes convenient to characterize the field by using the generalized complex potential |V(z), for which, in view of (1.1), (1.3) we easily obtain

t i m I J(~)dxt. a l;

(4.3)

Example. As an illustration, consider the field of a doubly periodic system of metal (~=~) of square section (Fig.2). The field H0=e0nst is directed along the x axis. rods

The side of a square a=6; the basic periods are ~t=8, ~2=4+8i. When the domain S is discretized, the basic square is divided into 36 square elements.

The magnetization J is calculated from (2.6), using form (4.2) for ~gl and expansions (4.1) for the kernels.

In Fig.2 we show lines ~=c0nsL and $=c0nst, obtained after calculating the magnetiza- tion J with the aid of (4.3).

REFERENCES

i. KHARADLY M. and JACKSON,W., The properties of artifical dielectrics comprising arrays of conducting elements, Prec. IEE, iO0, No.65, 199-211, 1953.

2. DMITRIEV V.I., Electromagnetic fields in inhomogeneous media (Elektrkomagnitnye polya v neodnorodnykh sredakh), Izd-vo MGU, Moscow, 1969.

3. TOZONI O.V., The method of secondary sources in electrical engineering, Energiya, Moscow, 1975.

4. STRAKHOV V.N., On the theory of a plane inverse problem of gravimetry for the contact sur- face, Dokl. Akad. Nauk SSSR, 249, No.5, 1091-1095, 1979.

5. KNYAZ' A.I., Plane-parallel fields and boundary value problems of the theory of analytic functions, in: Cybernetics and computing (Kibernetika i vychisl, tekhn.), No.42, Naukova dumka, Kiev, 1978.

6. VEKUA I.N., Generalized analytic functions (Obobshchennye analiticheskie funtskii), Fizmatgiz, Moscow, 1959.

7. HURWITZ A. and COURANT R., Vorlesungen'~ber allgemeine Funktionentheorie und elliptische Funktionen, Springer, 1929.

Translated by D.E.B.