- Email: [email protected]
Asymptotic study of transverse conductivity in superconducting multifilamentary composites
hf. 1. Engng Sri Vol. 20. No. 4, pp. 587600. Printed in Great Britain.
W2~7!25/82/040?87-14103.0010 0 19R2 Pergamon Prers Ltd.
1982
ASYMPTOTIC STUDY OF TRANSVERSE CONDUCTIVITY IN SUPERCONDUCTING MULTIFILAMENTARY COMPOSITES A. MIRGAUX L.E.M.T.A., 2 rue de la Citadelle, B.P. 850, F 5401I Nancy Cedex, France and J. SAINT JEAN PAULIN Laboratoire d’Analyse Numtrique, UniversitC Paris VI, C.N.K.S., 4, Place Jussieu, F 75230Paris Cedex 05, France Abstract-This paper deals with superconducting multifilamentary composites with periodic structure in the presence of a weak electromagnetic field. We study the magnetic induction and establish an approximation result when the period is small (in the framework of homogenization theory). Then we determine some physical constants of an “equivalent” homogeneous conductor, and thus transverse conductivity can be obtained. For various proportions of superconductor, we compute both the values of transverse conductivity proposed by different physicists and the mathematical value obtained in this paper (we use finite elements and an overrelaxation method); we compare them between themselves and with recent experimental results; we note that our results are in very good agreement with Carr’s formula and with the experimental measures of Davoust.
INTRODUCTION AN IMPORTANT feature of superconductors is that under certain conditions their electric resistance is negligible when the temperature is inferior to a certain critical temperature T,. Their technological interest is due to the fact that they can carry important densities of electric current in the presence of high magnetic fields. Technical considerations lead to use superconducting multifilamentary composites @MC): the superconductor is divided into many very thin elementary fibres isolated by a matrix with high conductivity. The distribution of the fibres in the matrix is periodic. In a first approximation, we can consider that the matrix and the fibres are infinitely long cylinders. One of the main problems connected with SMC is the following: in the presence of a variable magnetic field, a loss of energy occurs in the SMC, due to the presence of dissipative currents in the matrix and, if it is too important, it may lead the superconductor back to normal state. This loss of energy can be described by means of a macroscopic transverse conductivity flT*
Recent experiments were made by Davoust: she submits the SMC to an electromagnetic field parallel or perpendicular to its axis and makes various series of measures to reach OT (and hence she deduces the loss of energy); in [ll she gives experimental results and recalls different values of uT determined by physicists[2-6]. The aim of this paper is to reach (TT from a mathematical point of view through homogenization methods and numerical calculation and compare these results both with experiments and with the theoretical values proposed by physicists. In Section 1, we start from Maxwell’s equations and derive the statement of the problem. We prove existence and uniqueness theorems for the magnetic induction. In Section 2, we solve the corresponding homogenization problem when the magnetic field is parallel to the axis of the SMC: we give the limit solution, the first correcting terms and error estimates. In Section 3, we determine physical constants for a homogeneous conductor having the same shape as the SMC and in which the magnetic induction is the limit solution obtained in Section 2 (the exterior magnetic field being unchanged). Thus we derive a formula for transverse conductivity. In Section 4, we compute the homogenized coefficients for various proportions of super587
588
A.MIRGAUXandJ.SAINTJEAN PAULIN
conductor. We use a finite element method and solve the approximated problem with an overrelaxation method. We calculate the transverse conductivity of the homogeneous conductor defined in Section 3. It appears then clearly that our results, the experimental measures made by Davoust and Carr’s formula agree perfectly, whereas Keller’s and Wilson’s formulae give somewhat different results. I.STATEMENTOFTHEPROBLEM-EXISTENCEANDUNICITYRESULTS We consider a cylindrical SMC with superconducting cylindrical fibres, both infinitely long, and we suppose it is submitted to a weak exterior uniform time-variable magnetic field parallel to its axis H,,,(t) = HT(t)k3 (with HI(O) = 0). Since this field is parallel to the axis of the SMC, it is not distorted outside the conductor, thus its value on the lateral boundary equals the given value H,,,(t) at infinity (i.e. far from the SMC) and we merely have to study it in the SMC. We denote by R, R*, R’ (i = 1, . . . , N) the cross-sections of the SMC, the matrix and the fibres; we suppose that two distinct fibres do not meet, and that fibres do not meet the boundary either. Moreover, we suppose that 0, 0*, R’ (i = 1, . . . , N) are open bounded subsets of I?’ with a regular boundary. We keep the usual notation for the electric field, the induction. . As a first approximation, we can write Maxwell’s equations in the form J - curl H = 0 s+curl
E=O
(1.1)
1 * infi*xRx(O,T).
(1.2)
div B = 0. ,
(1.3)
The material laws are in the case of weak fields (1.4) J=cr,,,E
I
in R* x R X (0, T). (1.5)
B=O
(1.6) in R’ x R X (0, t).
E=O
(1.7)
The transmission conditions are [B * n] = 0
(1.8) on aRxRx(0,
[Hxn]=O
T). (1.9)
i
[B . n] = 0 \
(1.10) on ani x R x (0, T). (1.11)
[Hxn]=Oj where [ ] denotes a jump and n is the unitary normal on JR exterior of a* x R. The data are H(t) = HT(t)k,
on &Ix R B(x, 0) = 0
x
R or afii
(with H?(O) = 0)
on R*.
x
R pointing to the
(1.12) (1.13)
Asymptotic study of transverse conductivity
589
Remark. In experiments the initial induction is zero; for results when B(x, 0) f 0 see [7]. The geometry of the system and the fact that data are independent of x3 lead us to consider the magnetic induction B as a function independent of x3. Moreover, we already know that B is zero in the fibres; hence we merely have to determine B(.q, x2, t) when (x1, xZ)E a*. Notation. In the following, Greek indices may only take the values 1 or 2, whereas Latin indices may take the values 1, 2 or 3. As usual, comma means derivation (e.g.Bi,i = (JBJJXj)). Using the fact that B is independent of x3, we can derive easily from (l.lt(1.13) a decoupled problem for the magnetic induction: we get a problem (Q,) in B3 alone, and a problem (QJ in which B, and B2 are coupled. Problem Q,
aB3 ’ at ~4~
---
2&=(-j an
B,=dT
B,(., 0) = 0
hB=O
3
on aRi x (0, T)
(a/an
in s1* X (0, T)
(1.14)
denotes normal derivative)
(1.15)
on aR x (0, T)
(1.16)
on R*
(1.17)
Problem Q2
aB
n_-
‘AB=O
in R* X (0, T)
(1.18)
4, = 0
in R* x (0, T)
(1.19)
B, = 0
on aR x (0, T)
(1.20)
B,n, = 0
on ail x (0, T)
(1.21)
B,,z - Bw = 0
on alli x (0, T)
1.22)
B,(., 0) = 0
on a*.
1.23)
at ffmpo a
The existence and uniqueness results are given in the following proposition. Proposition 1.1. We make the former regularity assumptions and we suppose that HT E L*(O, T). Then problem (Q,, Q2) has a unique solution such that B, E L* (0, T; H*(n*)) B3 E L*(O, T; H’(R*)) II L”(0, T; L*@*)) Moreover B, =O. Proof. Problem Q, is a classical parabolic problem and zero is trivially a solution of problem Q2. Hence the only point to prove is the uniqueness of the solution of problem Q2. We note that (1.18H1.23) is not a classical problem at all: condition (1.22) does not derive “naturally” from a variational formulation; uniqueness is proved by using the vector potential. Indeed, since B, satisfies (1.19)-(1.21), we can apply the results of [8]: there exists a vector potential A such that
(1.24)
aLO
xUES Vol.20.No.4-G
on aR* x (0, T)
(ala7 denotes the tangential derivative).
A. MIRGAUX and J. SAINT JEAN PAULIN
590
Using the regularity assumptions, we see that if B, is a solution of problem Q2, then A is a solution of 1
aA
z-oA-4
aAz-
in R* x (0, T)
=gW
m
0
(1.26) . on all x (0, T)
aLO
an-
(1.27)
i
aLO
(1.28)
-z-
on ati x (0, T) A(., t) = A(., 0) +[
g(s) ds
A(., 0) = c
(1.29) on CR*
(1.30)
where g(t) is a function of t alone and c is a constant (the interesting point is that the troublesome condition (1.22) is turned into a much more agreeable one (1.29)). Classical results give the existence and uniqueness of a solution to problem (1.25) (1.27) (1.29) (1.30), moreover A(x,t)=
c+
in Q* x (0, T)
‘g(s)ds I0
(1.31)
is a solution of (1.25) (1.27) (1.29) (1.30), hence it is the only one; it also verifies (1.26) (1.28). We deduce from (1.24) (1.31) that B, is zero, q.e.d. Remark. In the general case (i.e. when the exterior magnetic field is not parallel to the axis of the SMC) we can also prove existence and unicity results but by a different method, using results of [8,9]; see [7] for details. The same method is also efficient to prove existence and unicity results when the fibres are infinitely long and twisted, which is a physically more realistic case than the model case treated here.
2. HOMOGENIZATION
When there are too many fibres, the solution of problem (Qr, f&) cannot be calculated directly by numerical methods, and hence the knowledge of an easily computable approximation (and of error estimates) is important. We consider a periodic configuration corresponding to the representative reference lY in the cross section, where Y = [0, I,[ x [0, 12[is the cell of reference in R2 and E is “small”. Notation. We keep the notations of problem (Q,) and (QJ, just adding the index E to show the dependence in E (e.g. B’ is the magnetic induction). We denote by F the part of Y corresponding to the cross section of the superconducting fibres. For simplicity, we suppose that F is a connected open subset of f, locally on a same side of its boundary (this is the case with the SMC studied by Davoust). We set y*= y-p
es = IF//( YI
(i.e. Y* corresponds to the matrix and es is the proportion of superconductor). We are interested in the behaviour of B, when e is small. Proposition 2.1. We make the same assumptions as in proposition 1.1 and we also suppose that HgO)=O,
aHTE LZ(OT) at 7 ’
q
at
E L2(0, T).
Asymptotic study of transverse conductivity
591
Then & - By in L2 (0, T; L2(Cl))weakly r+l where B$ is the solution of problem (Q*) defined by
a@
a2BT
in fi x (0, T)
(2.1)
Bf = (1 - #)&IT
on JR x (0, T)
(2.2)
BT(., 0) = 0
on R.
(2.3)
--
at
-o
%flax,ax,
The constants qas are proportional to (~Iu,,,,cL~) and will be determined later on. Proof. By means of the translation v,= B;-bHT
(2.4)
we transform (1.14),-(1.17), into a homogeneous boundary value problem:
au -Av~=-~~~ 1
inR$x(O,T)
A-
at
UmPO
du,=()
on a@ x (0, T)
v, = 0
on aSZx (0, T)
v,(., 0) = 0
on Q.
an
This can be considered as a Neumann problem in an open set with holes, and we can apply the methods and the results due to L. Tartar, and explained with details in [7, 10, 111.There exists an extension P,v, of v, to 0 such that P$, - v* r-b0
in L2(0, T; H@))
weakly (2.5)
in L2(0, T; H;(a)) weakly where v* is the solution of in R x (0, T) on aR x (0, T)
(2.6)
on R. The constants qaB are defined as follows. We first define w E H’( Y’), W(Y)={ I
w dy = 0, w Y-periodic Y’
I
(2.7)
Let xa be the solution of
-A(x”-ya)=O
in Y*
$ (x“ - y,) = 0 on aF (n is the unitary normal on aF).
(2.8)
A.MIRGAUXandJ.SAINTJEAN PALJLIN
592
The constants qoBare given by
We now turn back B;. Remembering that B; is zero in the superconducting fibres, we have
B; = xJ’&
($: characteristic function of fig).
(2.10)
We pass to the limit in (2.10) by using (2.4) (2.5) (2.6) and thus we get the accounted result. It is interesting to obtain a more precise estimate of B;. Since B; is known exactly in the fibres (it is zero there) it s&ices to obtain such an estimate in the matrix: Proposition 2.2. Under the assumptions of proposition 2.1, we have
I& - &lL”(O, T; L?rq)) s d2
(c indep. of e)
x aB3 s cJE’/2 (c’ indep. of E) - E 2% L*to,T:H’(s-Q)
0 II
B;-B,+(E/ya where B3 is the solution of
a&
__-
at B3
=
-_
a24
“_o
qaS axaaxB POW
ht.7 0) = 0
in s2 x (0, T) on JQ x (0, ‘i’) on 0.
Proof. The proof is based on the method of multiple scales (see [ 121).In a first step, a forma1 identification gives the first terms of the asymptotic expansion of Bj. The second step is a justification of the expansion established formally, by means convergence rest&s. For details see [7). 3.INTERPRETATIONOFTHEHOMOGENIZEDCOEFFICIENTS. DETERMINATIONOFTRANSVERSECONDUCTIViTY We proved in proposition 2.1 that when there are many fibres periodically distributed, the induction in the SMC can be approximated by B* = BTk3 (where B* is the solution of (2.1)-(2.3)). Now the question is: does there exist a homogeneous conductor of the same shape, for which the induction would also be B* (the exterior field being unchanged)? If so, we want to determine physical constants of such a conductor. In a first step we consider a homogeneous aniso~opic conductor C and we write a problem satisfied by the induction B. In a second step we use particular symmetry properties of the physical problem considered to simplify the problem obtained in the 1st step and we show that it has a unique solution B. In a third step, we establish that B = Bfjk3provided certain relations are satisfied between coefficients of the conductor C and the homogenized coefficients q-p We then consider this honogeneous conductor (which is not isotropic in R3) as equivalent to the SMC. 1st step. Magnetic induction in a homogeneous anisotropic conductor C Let Z (resp. FL)be the conductivity matrix (resp.: the permeability matrix); X and p are constant symmetric invertible matrices (see [13]). We introduce
s = (S$)= c-’
T = (tij) =
/L-‘*
(3.1)
We derive easily from Maxwell’s equations and from the comportment laws that the induction
Asymptotic study of transverse conductivity
593
B satisfies %
+
‘%jkElpqskl
fqrBr,pj
=
0
in Cx(0, T)
Bjnj = 0 on Cx(0, T) Eij,(tj&b& - 63jHT)n, Bj(.,
(3.2)
= 0
0) = 0
in C
(where cij& and Sij have the usual meaning?). 2nd step. Simplification of the former system when C is a cylinder. Existence and uniqueness of the solution Since the SMC is a cylinder, we expect that its global behavior resembles the one of a homogeneous orthotropic cylindrical conductor; this means that the matrices Z and p have the form (3.3)
(Hence the inverse matrices S and T have the same form). Moreover they have the following properties [ 131
I
%fl22-d2>0
u33
>
(3.5)
0
/41P22-/.42>0 I P33
>
(3.6)
0
Since the exterior field is uniform and parallel to the axis of the conductor, we seek the induction as a function independent of x3. Proposition 3.1. Suppose that C. and p have form (3.3) and (3.4) and satisfy (3.5) and (3.6). Suppose also that HT E L2 (0, T). Then the magnetic induction B in the homogeneous conductor C is a solution of the following decoupled system (where R is the cross-section of C). l?B -$ + l ,3e,g&,B.,~
=0
in R x (0, T)
B,n, = 0 flaBan - t,,B,n, = 0
Cjjh=
(3.8) on &I x (0, T)
+I if (i, j, k) is an even permutation of (1, 2, 3) -1 if (i, j, k) is an odd permutation of (1,2, 3) 0 if two indices are equal.
6,,= 1 if i=j 8, ( Oifi=j.
(3.7)
(3.9)
594
A. MIRGAUX and J. SAINT JEAN PAULIN B,(.,
0) = 0
in R
(3.10) (3.11)
t&4 = J-0’
on JR x (0, T)
(3.12)
B,(., 0) = 0
in R
(3.13)
The problem (3.7H3.13) has a unique solution B such that B, E L2 (0, T; H*(n)) B3 E L* (0, T; H’(R)) fI L”(0, T; L’(n)). Moreover B, = 0
(Y= 1,2.
Proof. The existence and uniqueness of B3 are easily deduced from the fact that (3.11) can be rewritten %
-
f33iS22B3.1,
-
2&2B3,12
+
@3,221
= 0
in R x (0, T)
(3.14)
(The coerciveness conditions are satisfied and (3.14) (3.12) (3.13) is thus a classical parabolic problem). The existence of a solution to (3.7)-(3.10) is trivial; uniqueness is proved by the same kind of argument as in proposition 1.1 (see [7] for details). 3rd step. Physical constants of an equivalent homogeneous conductor C such that B = BTk3 Theorem 3.2. We suppose that the matrices X and p characterizing the homogeneous conductor C satisfy (3.3H3.6) with CL33 =
1 uap = (1 - (%o
%I6 411q22-
(I-
@)/&I
(3.15)
(the qas are given by (2.9))
(3.16)
d2
Then the magnetic induction B in the homogeneous conductor C equals the “homogenized” induction B* = Bfk3 (the latter is the limit, in the sense of Proposition 2.1, of the induction B’ in the SMC). Proof. We saw that B3 (resp. B,) is characterized as the unique solution of (2.1)-(2.3) (resp. (3.11)-(3.13)). Therefore, B3 and B$ will be equal if the equations on 0 x (0, T) and the boundary conditions on a0 x (0, T) are identical, i.e. 411= f33S22
q12= - f33S12 q22= t33s11 (1 - @)/A0= t;: .
The last relation gives trivially (3.15); we prove (3.16) by an easy calculation using the explicit expression of uas (remember that Z = S-l). Remark. The expression (3.15) for parallel permeability is exactly the same as the one obtained by experiments (see [l]).
595
Asymptotic study of transverse conductivity
Corollary 3.3. We make the same assumptions as in Theorem 3.2. Moreover we suppose that the part F of the cell of reference Y corresponding to the superconducting fibres in symmetric with respect to the medians of Y. Then the matrix C is diagonal. Proof. It follows from (2.8) and Green’s formula that qea can be rewritten
The symmetry assumption implies that
Hence q12is zero and Corollary 3.3 is then a trivial consequence of formula (3.16).
4. NUMERICAL
COMPUTATION
We want to compute the physical constants umPof the equivalent homogeneous conductor described in Section 3. It is obvious from (3.16) and (2.9) that the main point is to calculate the solution x” of (2.8). In a first step we approach an equivalent formulation of problem (2.8) with a finite element method. In a second step we compute its solution with an overrelaxation method. In a third step we calculate the homogenized coefficients Q for various proportions of superconductor. In the end we determine the transverse conductivity oT and compare our results both with recent experimental measures and with values proposed by different physicists. 1st step. Finite element approximation
of an equivalent formulation
of problem
2.8
The solution X~ of problem (2.8) can also be defined by the equivalent minimization problem (see [141) xn is the unique function in W(Y) such that (4.1) where: (4.2) (n, is the component on the y, axis of the unitary normal on dF pointing to the exterior of Y*).
Since the cross-section of the fibres form a regular hexagonal distribution, we can take Y = [O,11X [O,(l/d/3) as the cell of reference (see Fig. 1). The region Y* is subdivided into triangular finite elements (T,+,)with normal conventions (see 11% 161). The spaces H’(Y) and W(Y) are, respectively, approached by U,, = {u E cO(Y*); u polynomial of degree 1 on each triangle} W,,=(w,, E U,,; wh Y-periodic,/y*w,,dy=O). Thus we obtain an interne approximation
(4.3) (4.4)
of problem (4.1)
,y; is the unique function in wh which minimizes the functional Jol(w,,) on W,,.
(4.5)
We now want to obtain an explicit formulation of the functional J,. We introduce the
A. MIRGAUX and J. SAINT JEAN PAULIN
596
Fig. 1. Cross-section
of the SMC. The shaded hexaeones corresoond to the superconducting dotted line to the bcundarybf EY.
fibres and the
following notations: ( ThF)= {TF E Th; a side of T, belong to JF} IF = length of this side (&,, MF2) Vh= { uh E U, ; u,, Y-periodic} & = characteristic function of the triangle T. Then we can write
where & is the linear affine function defined by 4T(Mi)
for i= 1,3
= uMi = Van
1 if Mi is a vertex of T. If we introduce ‘YT,= b2 - b3 &, = a3 - a2 YT, = 43 aT2= b3 - b, pTz= al - a3 yT2= ah aT3= b, - b2 PT3 = 02 - al YT,= aA-
a42 ah ah
where (ai, bi) are the coordinates of the node M;, we obtain the following explicit formulation of
J, J.(Q)=;
(46)
c
TE T,,
2nd step. Numerical solution We can state the approximated problem (4.6) into the equivalent form minimize Ja(uh) on vh
I
1
with the constraint:
I
y, v,, * dy = 0.
591
Asymptotic study of transverse conductivity
We minimize the functional 1, by using an over-relaxation method with optimal parameter. The Y* zero-mean condition is verified after each cycle of relaxation. Before describing the algorithm, we introduce the following simplified notations vi = vMi, Mi a node of Th
w;:= (WY,. . .
n+l/2 -(II;+‘, V h,i
)
WY,. . .
. . . , vy;,
)
wi)
vy+“2, WY+,,. . . , wy.J).
N is the number of variable nodes according with the Y-periodicity condition. For each step of a relation cycle (index: i), we determine at first v;+“~ such that
55(v;,;‘/2)
=
0
1
then we calculate vy+’ by vy+’ = (1 -
w)wYt 0 * v;+“2
where o is the optimal parameter (o = 1.7 in our case). When a relaxation cycle is over, we calculate the constant C”+’ such that wi+’ belongs to wh* n+l wh
_ -(v;+‘tf?+‘,
cn+1=_
1 3mdY*)
...
c Tech
)
vy+'tC"+', .. . )
mdT)[i
v;+'tC"+l)
0~~1 j=l
Remark. ,y(Iwas calculated with SOLAR-16 and IRIS-80 computers in simple precision; the overrelaxation method is stopped when
3rd step. Calculation of the homogenized coefficients Now that Problem (4.1) is solved we can calculate the homogenized coefficients by 1 fl&oQ3 = 4:s = mes ( y*) Y*V(X~- Y,) . Vx’ - YS)dy I for various proportions of superconductor. The results are given in Table 1. Table 1. Calculation of the homogenized coefficients (Nombre de de&
de IibertC= N - 1)
A. MIRGAUX and J. SAINTJEANPAULIN
598
We can observe that the diagonal coefficients are equal and the others are negligible according to the accuracy of the calculation. These results agree comp~eteIy with Corollary 3.3. We have taken for the numerical computations a. cell of reference Y in which the part F corresponding to the superconducting fibre is symmetric with respect to the medians of Y (see Fig. 1). 4th step. Transverse conductivity In Section 3, we have determined the conductivity matrix of the homogeneous conductor (3.16). With the hypothesis of Corollary 3.3, we derive the following formula for the transverse conductivity
:=(l-iP)y
(where 9 = & = &).
For various proportions of superconductor we calculate the ratio udu” and we compare these results both with recent experimental measures (see [l]) and with values obtained from some formulae proposed by different physicists. In the following we give indication about the formulae, we produce in Table 2 all the interesting values and we draw on Fig. 2 the corresponding curves. Some physicists have in fact estimated the transverse conductivity ffT by a local analysis of the different electromagnetic variables. They suppose that the supraconducting filaments have circular cross section (diameter: d). We denote by u the distance between the axis of the filaments, Keller141 considers a composite medium containing a square array of filaments. Laplace’s equation applied at this configuration leads to the formula
Later Wilson studies the twisted multifilament superconductor
and proposes an ap-
Table 2. Numerical comparison of the different formulae for transverse conductivity (with the notation y = ~(2i?s~3/~)) 8s
0.028
0.111
0.250
0.444
0.563
0.694
f formula obtained by the hcmcqenization method "PC__!___ Orn
l-240
1.658
0.840
I
/ 1.048
0.766
/
j
/
2.629
3.618
5.582
7,624
11.569
2.597
3.577
5,536
7.547
11.500
2.105
3.333
4.717
8.000
12.349
26.316
1.056
2.228
6.685
1 10.837
/ 24.175
Cl-B%l
CARS'S f0naula % --7 Orn
d
I.058
1-c
1.250
1
1.667
I
j
1st WILSON’S formula -,L % 0 m
2nd WILSMI'S 1Ir,l_L Orn
1.214
1.538
0.204
0.514
1-y
formula t
/
3.549
I
1 1-y -
1.200
1.499
2.000
3.000
4.000
6.300
8.000
12.048
2.447
2.755
3.223
4.056
4.825
6.285
7,605
11.396
599
Asymptotic study of transverse conductivity CIT global transverse conductivity OH
conductivity of the matrix
1
a0
a6
.on of the formula obtained by the homogenization method
0
experimental measure
-cARR'
formula irst WILSON'S formula
----2nd
-*.. -___
WILSON'S formula
_ 3rd WILSON'S formula -KELLER'S
formula
Fig. 2. Graphical comparison of the different results for transverse conductivity,
proximation in two dimensions by a “sloping sheet”. Without taking the real local distribution of the currents in the matrix into account, he obtains the two formulae z==,-r-
1
CT 3 y -=-I) alll 7r l-y
when the filaments are distant enough[5] when the filaments are near to each other[6].
Remark. In our cell of reference we use hexagonal cross section filaments (see Fig. 1). To use the precedent formulae we must obtain a vafue of y corresponding to equivalent circular cross section fibres. If we take circular filament with the same surface we have
but if the circle is only interiorly tangent with the hexagone, we obtain: y = +/(#). More recently Carr[2,3] makes the following analogy. He supposes that the filament are diamagnetic parallei circular cylinders in the vacuum. The calculus of the Lorentz’s field with the associated Laplace’s equation allows to join the apparent conductivity with the ratio 0’. So we obtain the formula 1+ BS z=m
600
A. MIRGAUX and J. SAINT JEAN PAULIN
We note the excellent accordance of our results with Carr’s formula and with the experimental measures of Davoust. Acknowledgement-We wish to thank Prof. A. Mailfert (Laboratoire d’Electronique et d’Electrotechnique Automatique, Institut National Polytechnique de Lorraine) who proposed this problem; we are grateful to him for several fruitful discussions particularly about the physical aspects of the subject.
REFERENCES [I] M. E. DAVOUST, Conductiviti ilectrique Transverse darts /es Fils de Materiaux Composites Multifilamentaires Supraconducteurs. These de 3tme cycle, Paris (1978). 121W. J. CARR, Jr, Phys. Rev. B 11(4), 1547(1975). [3] W. J. CARR, Jr, J. Appt. fhys. 46(9), 4043 (1975). [4] J. B. KELLER, J. Appl. Phys. 34(4), 991 (1963). [S] M. N. WILSON, J. D. LEWIN, P. F. SMITH, A. H. SPURWAY and C. R. WALTERS, 1. Phys. D, Appl. Phys. 3(11), 1517 (1970). [6] M. N. WILSON, Applied Superconductivity Conference, Annapolis. IEEEE Transactions of Magnetics (1972). [7] J. SAINT JEAN PAULIN, Etude de Quelques Problemes de Mecanique et d’Electrotechnique lies aux Methodes d’Hotnogheisation. These, Paris (1981). [E] 0. A. LADYZHENSKAYA, The Mathematical Theory of Viscous Incompressible Flow. Gordon Breach, New York
(1963). [9] G. DUVAUT and J. L. LIONS, Les Inequations en Mecanique et en Physique. Dunod, Paris (1972). [IO] D. CIORANESCU and J. SAINT JEAN PAULIN, CR. Acad. Sci. Paris Strie A, UM,899 (1978). [I I] D. CIORANESCU and J. SAINT JEAN PAULIN, J. Math. Anal. Appl. 71,590 (1979). 112) A. BENSOUSSAN, J. L. LIONS and G. PAPANICOLAOU, Asymptotic Methods in Periodic Structures. North Holland, Amsterdam (1978). [ 131L. LANDAU and E. LIFCHITZ, Electrodynamique des Milieux Continus. Editions MIR, Moscou (1969). [14] R. TEMAM, Numerical Analysis. D. Reidel, Dordrecht, Holland (1981). [IS] R. GLOWINSKI, J. L. LIONS and R. TREMOLIERES, Analyse Numerique des Inequations Variationnelles. Tome I, Dunod, Paris (1976). [I61 J. CEA, R. GLOWINSKI and J. C. NEDELEC, Application des Methodes d’optimisation de Differences et d’l%ments Finis a 1’Analyse Numirique de la Torsion Elastoplastique dune barre Cytindrique. Cahier no 12 de I’Iria, France (1974). (Received 10 April 1981)
W2~7!25/82/040?87-14103.0010 0 19R2 Pergamon Prers Ltd.
1982
ASYMPTOTIC STUDY OF TRANSVERSE CONDUCTIVITY IN SUPERCONDUCTING MULTIFILAMENTARY COMPOSITES A. MIRGAUX L.E.M.T.A., 2 rue de la Citadelle, B.P. 850, F 5401I Nancy Cedex, France and J. SAINT JEAN PAULIN Laboratoire d’Analyse Numtrique, UniversitC Paris VI, C.N.K.S., 4, Place Jussieu, F 75230Paris Cedex 05, France Abstract-This paper deals with superconducting multifilamentary composites with periodic structure in the presence of a weak electromagnetic field. We study the magnetic induction and establish an approximation result when the period is small (in the framework of homogenization theory). Then we determine some physical constants of an “equivalent” homogeneous conductor, and thus transverse conductivity can be obtained. For various proportions of superconductor, we compute both the values of transverse conductivity proposed by different physicists and the mathematical value obtained in this paper (we use finite elements and an overrelaxation method); we compare them between themselves and with recent experimental results; we note that our results are in very good agreement with Carr’s formula and with the experimental measures of Davoust.
INTRODUCTION AN IMPORTANT feature of superconductors is that under certain conditions their electric resistance is negligible when the temperature is inferior to a certain critical temperature T,. Their technological interest is due to the fact that they can carry important densities of electric current in the presence of high magnetic fields. Technical considerations lead to use superconducting multifilamentary composites @MC): the superconductor is divided into many very thin elementary fibres isolated by a matrix with high conductivity. The distribution of the fibres in the matrix is periodic. In a first approximation, we can consider that the matrix and the fibres are infinitely long cylinders. One of the main problems connected with SMC is the following: in the presence of a variable magnetic field, a loss of energy occurs in the SMC, due to the presence of dissipative currents in the matrix and, if it is too important, it may lead the superconductor back to normal state. This loss of energy can be described by means of a macroscopic transverse conductivity flT*
Recent experiments were made by Davoust: she submits the SMC to an electromagnetic field parallel or perpendicular to its axis and makes various series of measures to reach OT (and hence she deduces the loss of energy); in [ll she gives experimental results and recalls different values of uT determined by physicists[2-6]. The aim of this paper is to reach (TT from a mathematical point of view through homogenization methods and numerical calculation and compare these results both with experiments and with the theoretical values proposed by physicists. In Section 1, we start from Maxwell’s equations and derive the statement of the problem. We prove existence and uniqueness theorems for the magnetic induction. In Section 2, we solve the corresponding homogenization problem when the magnetic field is parallel to the axis of the SMC: we give the limit solution, the first correcting terms and error estimates. In Section 3, we determine physical constants for a homogeneous conductor having the same shape as the SMC and in which the magnetic induction is the limit solution obtained in Section 2 (the exterior magnetic field being unchanged). Thus we derive a formula for transverse conductivity. In Section 4, we compute the homogenized coefficients for various proportions of super587
588
A.MIRGAUXandJ.SAINTJEAN PAULIN
conductor. We use a finite element method and solve the approximated problem with an overrelaxation method. We calculate the transverse conductivity of the homogeneous conductor defined in Section 3. It appears then clearly that our results, the experimental measures made by Davoust and Carr’s formula agree perfectly, whereas Keller’s and Wilson’s formulae give somewhat different results. I.STATEMENTOFTHEPROBLEM-EXISTENCEANDUNICITYRESULTS We consider a cylindrical SMC with superconducting cylindrical fibres, both infinitely long, and we suppose it is submitted to a weak exterior uniform time-variable magnetic field parallel to its axis H,,,(t) = HT(t)k3 (with HI(O) = 0). Since this field is parallel to the axis of the SMC, it is not distorted outside the conductor, thus its value on the lateral boundary equals the given value H,,,(t) at infinity (i.e. far from the SMC) and we merely have to study it in the SMC. We denote by R, R*, R’ (i = 1, . . . , N) the cross-sections of the SMC, the matrix and the fibres; we suppose that two distinct fibres do not meet, and that fibres do not meet the boundary either. Moreover, we suppose that 0, 0*, R’ (i = 1, . . . , N) are open bounded subsets of I?’ with a regular boundary. We keep the usual notation for the electric field, the induction. . As a first approximation, we can write Maxwell’s equations in the form J - curl H = 0 s+curl
E=O
(1.1)
1 * infi*xRx(O,T).
(1.2)
div B = 0. ,
(1.3)
The material laws are in the case of weak fields (1.4) J=cr,,,E
I
in R* x R X (0, T). (1.5)
B=O
(1.6) in R’ x R X (0, t).
E=O
(1.7)
The transmission conditions are [B * n] = 0
(1.8) on aRxRx(0,
[Hxn]=O
T). (1.9)
i
[B . n] = 0 \
(1.10) on ani x R x (0, T). (1.11)
[Hxn]=Oj where [ ] denotes a jump and n is the unitary normal on JR exterior of a* x R. The data are H(t) = HT(t)k,
on &Ix R B(x, 0) = 0
x
R or afii
(with H?(O) = 0)
on R*.
x
R pointing to the
(1.12) (1.13)
Asymptotic study of transverse conductivity
589
Remark. In experiments the initial induction is zero; for results when B(x, 0) f 0 see [7]. The geometry of the system and the fact that data are independent of x3 lead us to consider the magnetic induction B as a function independent of x3. Moreover, we already know that B is zero in the fibres; hence we merely have to determine B(.q, x2, t) when (x1, xZ)E a*. Notation. In the following, Greek indices may only take the values 1 or 2, whereas Latin indices may take the values 1, 2 or 3. As usual, comma means derivation (e.g.Bi,i = (JBJJXj)). Using the fact that B is independent of x3, we can derive easily from (l.lt(1.13) a decoupled problem for the magnetic induction: we get a problem (Q,) in B3 alone, and a problem (QJ in which B, and B2 are coupled. Problem Q,
aB3 ’ at ~4~
---
2&=(-j an
B,=dT
B,(., 0) = 0
hB=O
3
on aRi x (0, T)
(a/an
in s1* X (0, T)
(1.14)
denotes normal derivative)
(1.15)
on aR x (0, T)
(1.16)
on R*
(1.17)
Problem Q2
aB
n_-
‘AB=O
in R* X (0, T)
(1.18)
4, = 0
in R* x (0, T)
(1.19)
B, = 0
on aR x (0, T)
(1.20)
B,n, = 0
on ail x (0, T)
(1.21)
B,,z - Bw = 0
on alli x (0, T)
1.22)
B,(., 0) = 0
on a*.
1.23)
at ffmpo a
The existence and uniqueness results are given in the following proposition. Proposition 1.1. We make the former regularity assumptions and we suppose that HT E L*(O, T). Then problem (Q,, Q2) has a unique solution such that B, E L* (0, T; H*(n*)) B3 E L*(O, T; H’(R*)) II L”(0, T; L*@*)) Moreover B, =O. Proof. Problem Q, is a classical parabolic problem and zero is trivially a solution of problem Q2. Hence the only point to prove is the uniqueness of the solution of problem Q2. We note that (1.18H1.23) is not a classical problem at all: condition (1.22) does not derive “naturally” from a variational formulation; uniqueness is proved by using the vector potential. Indeed, since B, satisfies (1.19)-(1.21), we can apply the results of [8]: there exists a vector potential A such that
(1.24)
aLO
xUES Vol.20.No.4-G
on aR* x (0, T)
(ala7 denotes the tangential derivative).
A. MIRGAUX and J. SAINT JEAN PAULIN
590
Using the regularity assumptions, we see that if B, is a solution of problem Q2, then A is a solution of 1
aA
z-oA-4
aAz-
in R* x (0, T)
=gW
m
0
(1.26) . on all x (0, T)
aLO
an-
(1.27)
i
aLO
(1.28)
-z-
on ati x (0, T) A(., t) = A(., 0) +[
g(s) ds
A(., 0) = c
(1.29) on CR*
(1.30)
where g(t) is a function of t alone and c is a constant (the interesting point is that the troublesome condition (1.22) is turned into a much more agreeable one (1.29)). Classical results give the existence and uniqueness of a solution to problem (1.25) (1.27) (1.29) (1.30), moreover A(x,t)=
c+
in Q* x (0, T)
‘g(s)ds I0
(1.31)
is a solution of (1.25) (1.27) (1.29) (1.30), hence it is the only one; it also verifies (1.26) (1.28). We deduce from (1.24) (1.31) that B, is zero, q.e.d. Remark. In the general case (i.e. when the exterior magnetic field is not parallel to the axis of the SMC) we can also prove existence and unicity results but by a different method, using results of [8,9]; see [7] for details. The same method is also efficient to prove existence and unicity results when the fibres are infinitely long and twisted, which is a physically more realistic case than the model case treated here.
2. HOMOGENIZATION
When there are too many fibres, the solution of problem (Qr, f&) cannot be calculated directly by numerical methods, and hence the knowledge of an easily computable approximation (and of error estimates) is important. We consider a periodic configuration corresponding to the representative reference lY in the cross section, where Y = [0, I,[ x [0, 12[is the cell of reference in R2 and E is “small”. Notation. We keep the notations of problem (Q,) and (QJ, just adding the index E to show the dependence in E (e.g. B’ is the magnetic induction). We denote by F the part of Y corresponding to the cross section of the superconducting fibres. For simplicity, we suppose that F is a connected open subset of f, locally on a same side of its boundary (this is the case with the SMC studied by Davoust). We set y*= y-p
es = IF//( YI
(i.e. Y* corresponds to the matrix and es is the proportion of superconductor). We are interested in the behaviour of B, when e is small. Proposition 2.1. We make the same assumptions as in proposition 1.1 and we also suppose that HgO)=O,
aHTE LZ(OT) at 7 ’
q
at
E L2(0, T).
Asymptotic study of transverse conductivity
591
Then & - By in L2 (0, T; L2(Cl))weakly r+l where B$ is the solution of problem (Q*) defined by
a@
a2BT
in fi x (0, T)
(2.1)
Bf = (1 - #)&IT
on JR x (0, T)
(2.2)
BT(., 0) = 0
on R.
(2.3)
--
at
-o
%flax,ax,
The constants qas are proportional to (~Iu,,,,cL~) and will be determined later on. Proof. By means of the translation v,= B;-bHT
(2.4)
we transform (1.14),-(1.17), into a homogeneous boundary value problem:
au -Av~=-~~~ 1
inR$x(O,T)
A-
at
UmPO
du,=()
on a@ x (0, T)
v, = 0
on aSZx (0, T)
v,(., 0) = 0
on Q.
an
This can be considered as a Neumann problem in an open set with holes, and we can apply the methods and the results due to L. Tartar, and explained with details in [7, 10, 111.There exists an extension P,v, of v, to 0 such that P$, - v* r-b0
in L2(0, T; H@))
weakly (2.5)
in L2(0, T; H;(a)) weakly where v* is the solution of in R x (0, T) on aR x (0, T)
(2.6)
on R. The constants qaB are defined as follows. We first define w E H’( Y’), W(Y)={ I
w dy = 0, w Y-periodic Y’
I
(2.7)
Let xa be the solution of
-A(x”-ya)=O
in Y*
$ (x“ - y,) = 0 on aF (n is the unitary normal on aF).
(2.8)
A.MIRGAUXandJ.SAINTJEAN PALJLIN
592
The constants qoBare given by
We now turn back B;. Remembering that B; is zero in the superconducting fibres, we have
B; = xJ’&
($: characteristic function of fig).
(2.10)
We pass to the limit in (2.10) by using (2.4) (2.5) (2.6) and thus we get the accounted result. It is interesting to obtain a more precise estimate of B;. Since B; is known exactly in the fibres (it is zero there) it s&ices to obtain such an estimate in the matrix: Proposition 2.2. Under the assumptions of proposition 2.1, we have
I& - &lL”(O, T; L?rq)) s d2
(c indep. of e)
x aB3 s cJE’/2 (c’ indep. of E) - E 2% L*to,T:H’(s-Q)
0 II
B;-B,+(E/ya where B3 is the solution of
a&
__-
at B3
=
-_
a24
“_o
qaS axaaxB POW
ht.7 0) = 0
in s2 x (0, T) on JQ x (0, ‘i’) on 0.
Proof. The proof is based on the method of multiple scales (see [ 121).In a first step, a forma1 identification gives the first terms of the asymptotic expansion of Bj. The second step is a justification of the expansion established formally, by means convergence rest&s. For details see [7). 3.INTERPRETATIONOFTHEHOMOGENIZEDCOEFFICIENTS. DETERMINATIONOFTRANSVERSECONDUCTIViTY We proved in proposition 2.1 that when there are many fibres periodically distributed, the induction in the SMC can be approximated by B* = BTk3 (where B* is the solution of (2.1)-(2.3)). Now the question is: does there exist a homogeneous conductor of the same shape, for which the induction would also be B* (the exterior field being unchanged)? If so, we want to determine physical constants of such a conductor. In a first step we consider a homogeneous aniso~opic conductor C and we write a problem satisfied by the induction B. In a second step we use particular symmetry properties of the physical problem considered to simplify the problem obtained in the 1st step and we show that it has a unique solution B. In a third step, we establish that B = Bfjk3provided certain relations are satisfied between coefficients of the conductor C and the homogenized coefficients q-p We then consider this honogeneous conductor (which is not isotropic in R3) as equivalent to the SMC. 1st step. Magnetic induction in a homogeneous anisotropic conductor C Let Z (resp. FL)be the conductivity matrix (resp.: the permeability matrix); X and p are constant symmetric invertible matrices (see [13]). We introduce
s = (S$)= c-’
T = (tij) =
/L-‘*
(3.1)
We derive easily from Maxwell’s equations and from the comportment laws that the induction
Asymptotic study of transverse conductivity
593
B satisfies %
+
‘%jkElpqskl
fqrBr,pj
=
0
in Cx(0, T)
Bjnj = 0 on Cx(0, T) Eij,(tj&b& - 63jHT)n, Bj(.,
(3.2)
= 0
0) = 0
in C
(where cij& and Sij have the usual meaning?). 2nd step. Simplification of the former system when C is a cylinder. Existence and uniqueness of the solution Since the SMC is a cylinder, we expect that its global behavior resembles the one of a homogeneous orthotropic cylindrical conductor; this means that the matrices Z and p have the form (3.3)
(Hence the inverse matrices S and T have the same form). Moreover they have the following properties [ 131
I
%fl22-d2>0
u33
>
(3.5)
0
/41P22-/.42>0 I P33
>
(3.6)
0
Since the exterior field is uniform and parallel to the axis of the conductor, we seek the induction as a function independent of x3. Proposition 3.1. Suppose that C. and p have form (3.3) and (3.4) and satisfy (3.5) and (3.6). Suppose also that HT E L2 (0, T). Then the magnetic induction B in the homogeneous conductor C is a solution of the following decoupled system (where R is the cross-section of C). l?B -$ + l ,3e,g&,B.,~
=0
in R x (0, T)
B,n, = 0 flaBan - t,,B,n, = 0
Cjjh=
(3.8) on &I x (0, T)
+I if (i, j, k) is an even permutation of (1, 2, 3) -1 if (i, j, k) is an odd permutation of (1,2, 3) 0 if two indices are equal.
6,,= 1 if i=j 8, ( Oifi=j.
(3.7)
(3.9)
594
A. MIRGAUX and J. SAINT JEAN PAULIN B,(.,
0) = 0
in R
(3.10) (3.11)
t&4 = J-0’
on JR x (0, T)
(3.12)
B,(., 0) = 0
in R
(3.13)
The problem (3.7H3.13) has a unique solution B such that B, E L2 (0, T; H*(n)) B3 E L* (0, T; H’(R)) fI L”(0, T; L’(n)). Moreover B, = 0
(Y= 1,2.
Proof. The existence and uniqueness of B3 are easily deduced from the fact that (3.11) can be rewritten %
-
f33iS22B3.1,
-
2&2B3,12
+
@3,221
= 0
in R x (0, T)
(3.14)
(The coerciveness conditions are satisfied and (3.14) (3.12) (3.13) is thus a classical parabolic problem). The existence of a solution to (3.7)-(3.10) is trivial; uniqueness is proved by the same kind of argument as in proposition 1.1 (see [7] for details). 3rd step. Physical constants of an equivalent homogeneous conductor C such that B = BTk3 Theorem 3.2. We suppose that the matrices X and p characterizing the homogeneous conductor C satisfy (3.3H3.6) with CL33 =
1 uap = (1 - (%o
%I6 411q22-
(I-
@)/&I
(3.15)
(the qas are given by (2.9))
(3.16)
d2
Then the magnetic induction B in the homogeneous conductor C equals the “homogenized” induction B* = Bfk3 (the latter is the limit, in the sense of Proposition 2.1, of the induction B’ in the SMC). Proof. We saw that B3 (resp. B,) is characterized as the unique solution of (2.1)-(2.3) (resp. (3.11)-(3.13)). Therefore, B3 and B$ will be equal if the equations on 0 x (0, T) and the boundary conditions on a0 x (0, T) are identical, i.e. 411= f33S22
q12= - f33S12 q22= t33s11 (1 - @)/A0= t;: .
The last relation gives trivially (3.15); we prove (3.16) by an easy calculation using the explicit expression of uas (remember that Z = S-l). Remark. The expression (3.15) for parallel permeability is exactly the same as the one obtained by experiments (see [l]).
595
Asymptotic study of transverse conductivity
Corollary 3.3. We make the same assumptions as in Theorem 3.2. Moreover we suppose that the part F of the cell of reference Y corresponding to the superconducting fibres in symmetric with respect to the medians of Y. Then the matrix C is diagonal. Proof. It follows from (2.8) and Green’s formula that qea can be rewritten
The symmetry assumption implies that
Hence q12is zero and Corollary 3.3 is then a trivial consequence of formula (3.16).
4. NUMERICAL
COMPUTATION
We want to compute the physical constants umPof the equivalent homogeneous conductor described in Section 3. It is obvious from (3.16) and (2.9) that the main point is to calculate the solution x” of (2.8). In a first step we approach an equivalent formulation of problem (2.8) with a finite element method. In a second step we compute its solution with an overrelaxation method. In a third step we calculate the homogenized coefficients Q for various proportions of superconductor. In the end we determine the transverse conductivity oT and compare our results both with recent experimental measures and with values proposed by different physicists. 1st step. Finite element approximation
of an equivalent formulation
of problem
2.8
The solution X~ of problem (2.8) can also be defined by the equivalent minimization problem (see [141) xn is the unique function in W(Y) such that (4.1) where: (4.2) (n, is the component on the y, axis of the unitary normal on dF pointing to the exterior of Y*).
Since the cross-section of the fibres form a regular hexagonal distribution, we can take Y = [O,11X [O,(l/d/3) as the cell of reference (see Fig. 1). The region Y* is subdivided into triangular finite elements (T,+,)with normal conventions (see 11% 161). The spaces H’(Y) and W(Y) are, respectively, approached by U,, = {u E cO(Y*); u polynomial of degree 1 on each triangle} W,,=(w,, E U,,; wh Y-periodic,/y*w,,dy=O). Thus we obtain an interne approximation
(4.3) (4.4)
of problem (4.1)
,y; is the unique function in wh which minimizes the functional Jol(w,,) on W,,.
(4.5)
We now want to obtain an explicit formulation of the functional J,. We introduce the
A. MIRGAUX and J. SAINT JEAN PAULIN
596
Fig. 1. Cross-section
of the SMC. The shaded hexaeones corresoond to the superconducting dotted line to the bcundarybf EY.
fibres and the
following notations: ( ThF)= {TF E Th; a side of T, belong to JF} IF = length of this side (&,, MF2) Vh= { uh E U, ; u,, Y-periodic} & = characteristic function of the triangle T. Then we can write
where & is the linear affine function defined by 4T(Mi)
for i= 1,3
= uMi = Van
1 if Mi is a vertex of T. If we introduce ‘YT,= b2 - b3 &, = a3 - a2 YT, = 43 aT2= b3 - b, pTz= al - a3 yT2= ah aT3= b, - b2 PT3 = 02 - al YT,= aA-
a42 ah ah
where (ai, bi) are the coordinates of the node M;, we obtain the following explicit formulation of
J, J.(Q)=;
(46)
c
TE T,,
2nd step. Numerical solution We can state the approximated problem (4.6) into the equivalent form minimize Ja(uh) on vh
I
1
with the constraint:
I
y, v,, * dy = 0.
591
Asymptotic study of transverse conductivity
We minimize the functional 1, by using an over-relaxation method with optimal parameter. The Y* zero-mean condition is verified after each cycle of relaxation. Before describing the algorithm, we introduce the following simplified notations vi = vMi, Mi a node of Th
w;:= (WY,. . .
n+l/2 -(II;+‘, V h,i
)
WY,. . .
. . . , vy;,
)
wi)
vy+“2, WY+,,. . . , wy.J).
N is the number of variable nodes according with the Y-periodicity condition. For each step of a relation cycle (index: i), we determine at first v;+“~ such that
55(v;,;‘/2)
=
0
1
then we calculate vy+’ by vy+’ = (1 -
w)wYt 0 * v;+“2
where o is the optimal parameter (o = 1.7 in our case). When a relaxation cycle is over, we calculate the constant C”+’ such that wi+’ belongs to wh* n+l wh
_ -(v;+‘tf?+‘,
cn+1=_
1 3mdY*)
...
c Tech
)
vy+'tC"+', .. . )
mdT)[i
v;+'tC"+l)
0~~1 j=l
Remark. ,y(Iwas calculated with SOLAR-16 and IRIS-80 computers in simple precision; the overrelaxation method is stopped when
3rd step. Calculation of the homogenized coefficients Now that Problem (4.1) is solved we can calculate the homogenized coefficients by 1 fl&oQ3 = 4:s = mes ( y*) Y*V(X~- Y,) . Vx’ - YS)dy I for various proportions of superconductor. The results are given in Table 1. Table 1. Calculation of the homogenized coefficients (Nombre de de&
de IibertC= N - 1)
A. MIRGAUX and J. SAINTJEANPAULIN
598
We can observe that the diagonal coefficients are equal and the others are negligible according to the accuracy of the calculation. These results agree comp~eteIy with Corollary 3.3. We have taken for the numerical computations a. cell of reference Y in which the part F corresponding to the superconducting fibre is symmetric with respect to the medians of Y (see Fig. 1). 4th step. Transverse conductivity In Section 3, we have determined the conductivity matrix of the homogeneous conductor (3.16). With the hypothesis of Corollary 3.3, we derive the following formula for the transverse conductivity
:=(l-iP)y
(where 9 = & = &).
For various proportions of superconductor we calculate the ratio udu” and we compare these results both with recent experimental measures (see [l]) and with values obtained from some formulae proposed by different physicists. In the following we give indication about the formulae, we produce in Table 2 all the interesting values and we draw on Fig. 2 the corresponding curves. Some physicists have in fact estimated the transverse conductivity ffT by a local analysis of the different electromagnetic variables. They suppose that the supraconducting filaments have circular cross section (diameter: d). We denote by u the distance between the axis of the filaments, Keller141 considers a composite medium containing a square array of filaments. Laplace’s equation applied at this configuration leads to the formula
Later Wilson studies the twisted multifilament superconductor
and proposes an ap-
Table 2. Numerical comparison of the different formulae for transverse conductivity (with the notation y = ~(2i?s~3/~)) 8s
0.028
0.111
0.250
0.444
0.563
0.694
f formula obtained by the hcmcqenization method "PC__!___ Orn
l-240
1.658
0.840
I
/ 1.048
0.766
/
j
/
2.629
3.618
5.582
7,624
11.569
2.597
3.577
5,536
7.547
11.500
2.105
3.333
4.717
8.000
12.349
26.316
1.056
2.228
6.685
1 10.837
/ 24.175
Cl-B%l
CARS'S f0naula % --7 Orn
d
I.058
1-c
1.250
1
1.667
I
j
1st WILSON’S formula -,L % 0 m
2nd WILSMI'S 1Ir,l_L Orn
1.214
1.538
0.204
0.514
1-y
formula t
/
3.549
I
1 1-y -
1.200
1.499
2.000
3.000
4.000
6.300
8.000
12.048
2.447
2.755
3.223
4.056
4.825
6.285
7,605
11.396
599
Asymptotic study of transverse conductivity CIT global transverse conductivity OH
conductivity of the matrix
1
a0
a6
.on of the formula obtained by the homogenization method
0
experimental measure
-cARR'
formula irst WILSON'S formula
----2nd
-*.. -___
WILSON'S formula
_ 3rd WILSON'S formula -KELLER'S
formula
Fig. 2. Graphical comparison of the different results for transverse conductivity,
proximation in two dimensions by a “sloping sheet”. Without taking the real local distribution of the currents in the matrix into account, he obtains the two formulae z==,-r-
1
CT 3 y -=-I) alll 7r l-y
when the filaments are distant enough[5] when the filaments are near to each other[6].
Remark. In our cell of reference we use hexagonal cross section filaments (see Fig. 1). To use the precedent formulae we must obtain a vafue of y corresponding to equivalent circular cross section fibres. If we take circular filament with the same surface we have
but if the circle is only interiorly tangent with the hexagone, we obtain: y = +/(#). More recently Carr[2,3] makes the following analogy. He supposes that the filament are diamagnetic parallei circular cylinders in the vacuum. The calculus of the Lorentz’s field with the associated Laplace’s equation allows to join the apparent conductivity with the ratio 0’. So we obtain the formula 1+ BS z=m
600
A. MIRGAUX and J. SAINT JEAN PAULIN
We note the excellent accordance of our results with Carr’s formula and with the experimental measures of Davoust. Acknowledgement-We wish to thank Prof. A. Mailfert (Laboratoire d’Electronique et d’Electrotechnique Automatique, Institut National Polytechnique de Lorraine) who proposed this problem; we are grateful to him for several fruitful discussions particularly about the physical aspects of the subject.
REFERENCES [I] M. E. DAVOUST, Conductiviti ilectrique Transverse darts /es Fils de Materiaux Composites Multifilamentaires Supraconducteurs. These de 3tme cycle, Paris (1978). 121W. J. CARR, Jr, Phys. Rev. B 11(4), 1547(1975). [3] W. J. CARR, Jr, J. Appt. fhys. 46(9), 4043 (1975). [4] J. B. KELLER, J. Appl. Phys. 34(4), 991 (1963). [S] M. N. WILSON, J. D. LEWIN, P. F. SMITH, A. H. SPURWAY and C. R. WALTERS, 1. Phys. D, Appl. Phys. 3(11), 1517 (1970). [6] M. N. WILSON, Applied Superconductivity Conference, Annapolis. IEEEE Transactions of Magnetics (1972). [7] J. SAINT JEAN PAULIN, Etude de Quelques Problemes de Mecanique et d’Electrotechnique lies aux Methodes d’Hotnogheisation. These, Paris (1981). [E] 0. A. LADYZHENSKAYA, The Mathematical Theory of Viscous Incompressible Flow. Gordon Breach, New York
(1963). [9] G. DUVAUT and J. L. LIONS, Les Inequations en Mecanique et en Physique. Dunod, Paris (1972). [IO] D. CIORANESCU and J. SAINT JEAN PAULIN, CR. Acad. Sci. Paris Strie A, UM,899 (1978). [I I] D. CIORANESCU and J. SAINT JEAN PAULIN, J. Math. Anal. Appl. 71,590 (1979). 112) A. BENSOUSSAN, J. L. LIONS and G. PAPANICOLAOU, Asymptotic Methods in Periodic Structures. North Holland, Amsterdam (1978). [ 131L. LANDAU and E. LIFCHITZ, Electrodynamique des Milieux Continus. Editions MIR, Moscou (1969). [14] R. TEMAM, Numerical Analysis. D. Reidel, Dordrecht, Holland (1981). [IS] R. GLOWINSKI, J. L. LIONS and R. TREMOLIERES, Analyse Numerique des Inequations Variationnelles. Tome I, Dunod, Paris (1976). [I61 J. CEA, R. GLOWINSKI and J. C. NEDELEC, Application des Methodes d’optimisation de Differences et d’l%ments Finis a 1’Analyse Numirique de la Torsion Elastoplastique dune barre Cytindrique. Cahier no 12 de I’Iria, France (1974). (Received 10 April 1981)