Cluster method in composites and its convergence

Applied Mathematics Letters 77 (2018) 44–48

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Applied Mathematics Letters www.elsevier.com/locate/aml

Cluster method in composites and its convergence Vladimir Mityushev Institute of Computer Sciences, Pedagogical University, ul. Podchorazych 2, Krakow 30-084, Poland

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Article history: Received 3 September 2017 Received in revised form 2 October 2017 Accepted 2 October 2017 Available online 12 October 2017

abstract Extensions of Maxwell’s self-consistent approach from single- to n- inclusions problems lead to cluster methods applied to computation of the effective properties of composites. We describe applications of Maxwell’s formalism to finite clusters and explain the uncertainty arising when n tends to infinity by study of the corresponding conditionally convergent series. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Maxwell’s formalism Effective conductivity Cluster method

1. Introduction Maxwell’s self-consisting approach [1, page 365] is one of the most widely used methods for calculating the effective properties of composites. It is based on calculation of the field disturbed by a single inclusion embedded in a uniform host and further estimation of the macroscopic response via the dipole moments induced by many inclusions. A vast number of studies were initiated by this approach (see for instance [2–4] and works cited therein). Extensions of Maxwell’s approach from single- to n- inclusions problems are called cluster methods. From mathematical point of view, the cluster method means extension of the constructive solution to boundary value problems for a simply connected domain to a multiply connected domain and further averaging of the obtained local fields to estimate the effective constants. It differs from cluster analysis devoted to formation of clusters and to self-assembly studied as dynamical processes [5]. In the same time, the cluster method can resolve questions posed in cluster analysis, e.g. [6]. The obtained results can be summarized as follows: (i) Basing on the field around a finite cluster without clusters interactions one can deduce a formula for the effective conductivity only for dilute clusters. (ii) The uncertainty arising in various self-consisting cluster methods when the number of elements n in a cluster Cn tends to infinity is analyzed by means of the conditionally convergent series discussed in [7,8]. E-mail address: [email protected]. https://doi.org/10.1016/j.aml.2017.10.001 0893-9659/© 2017 Elsevier Ltd. All rights reserved.

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We consider the 2D conductivity problem with n circular inclusions solved exactly in [9] and derive the limit fromulae, as n → ∞. Discussion of this simple problem justifies necessity to consider the points (i) and (ii) during manipulations in the framework of Maxwell’s formalism. 2. Finite cluster ˆ = C ∪ {∞}. Consider Let z = x + iy denote a complex variable in the extended complex plane C non-overlapping disks |z − ak | < r (k = 1, 2, . . . , n), denoted below by Dk , of conductivity σ imbedded in the host material of the normalized unit conductivity occupying the domain D, the complement of all ˆ The potentials u(z) and uk (z) are harmonic in D and Dk , respectively, and the disks |z − ak | ≤ r to C. continuously differentiable in the closures of the considered domains except at infinity where u(z) ∼ x = Re z. The singularity of u(z) determine the external flux applied at infinity. The perfect contact condition (transmission problem [10]) between the components is expressed by two real relations [4] uk (z) = u(z),

σ

∂uk ∂u (z) = (z), ∂n ∂n

|z − ak | = r (k = 1, 2, . . . , n)

(1)

∂ where ∂n denotes the outward unit normal derivative to |z − ak | = r. Introduce the contrast parameter σ−1 ϱ = σ+1 . Two real equations (1) are reduced to the R−linear complex condition [4]

φ(z) = φk (z) − ϱφk (z),

|z − ak | = r (k = 1, 2, . . . , n)

(2)

where φ(z) and φk (z) are analytic in D and Dk , respectively, and continuously differentiable in the closures of the considered domains except at infinity where φ(z) ∼ z. The harmonic and analytic functions are related 2 by the equalities u(z) = Re φ(z) in D, uk (z) = σ+1 Re φk (z) in Dk . Consider the Schottky group of inversions and their compositions with respect to the circles |z − ak | = r, k = 1, 2, . . . , n (plus the identity element) ∗ z(k) =

r2 ∗ ∗ + ak , z(k := (z(k )∗ , 1 k2 ...,km ) 2 ...,km−1 ) k1 z − ak

(kj+1 ̸= kj ).

(3)

Exact solution of the considered problem for any |ϱ| < 1 was found in the form of the absolutely and uniformly convergent Poincar´e type series [9] φ(z) = z + ϱ

n ∑

∗ + ϱ2 z(k)

k=1

n ∑ ∑

∗ z(k + ϱ3 1 k)

k=1 k1 ̸=k

n ∑ ∑ ∑

∗ z(k + ··· 2 k1 k)

(4)

k=1 k1 ̸=k k2 ̸=k1

The set of inclusions Cn = ∪nk=1 Dk forms a finite cluster on the plane. Its important characteristic is the dipole moment M (n) [10] equal to the coefficient of φ(z) on z −1 which can be exactly written by (4). For our purposes it is sufficient to use the asymptotic formula M (n) = nϱr2 M(n) , (n)

M

=1−

(n) nϱr2 e2

(5) +

(n) n2 ϱ2 r4 e22

3 2 6

−n ϱ r

(n) [2e33

+

(n) ϱe222 ]

8

+ O(r ).

Here, the multiple sums arisen from (4) are shortly written as ∑ 1 ∑ 1 1 1 , e(n) pp = p+1 p p p n (ak − ak1 ) n (ak − ak1 ) (ak1 − ak2 )p k,k1 k,k1 ,k2 ∑ 1 1 = p+2 (p = 2, 3). n (ak − ak1 )p (ak1 − ak2 )p (ak2 − ak3 )p

e(n) p = e(n) ppp

k,k1 ,k2 ,k3

(6)

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For simplicity, we consider macroscopically isotropic composites when M (n) ∈ R. Following Maxwell’s formalism consider a large disk D of radius R0 containing N (R0 ) equal clusters C (n) . It is worth noting that n (number of disks per cluster) and N (R0 ) (number of clusters in D) are independent. Let the disk D (n) be occupied by a homogenized medium with an unknown effective conductivity σe . Its dipole moment is equated to the sum of the cluster moments [1] σe

(n)

−1

(n) σe

+1

R02 = N (R0 )M (n) .

(7)

The total number of small disks in the disk D is equal to nN (R0 ). Introduce the concentration of small disks in the plane f = nr2 limR0 →∞ N (R0 )R0−2 assuming that it exists. Of course, f does not depend on the local fraction of disks in the cluster. Substituting (5) into (7) we find σe(n) =

1 + f ϱM(n) + O(f 2 ). 1 − f ϱM(n)

(8)

Example. Let m be a natural number. Consider a square cluster consisting of n = (2m + 1)2 disks Dm1 m2 with the centers am1 m2 = m1 + im2 when m1,2 = −m, −m + 1, . . . , −1, 0, 1, . . . , m − 1, m. It follows from (n) (n) (n) (n) the symmetry and confirmed by computations that e2 = e22 = e222 = 0. The term α(n) = −2n3 e33 takes the values α(1) = 13.0, α(2) = 8.89, α(3) = 7.04, α(4) = 5.95, α(5) = 5.22, α(6) = 4.69, α(7) = 4.29, α(8) = 3.97 and tends to zero as n → ∞ (see Section 3). Therefore, the effective conductivities of these different composites, represented by clusters Cn , can be calculated by (8) with M(n) = 1 + α(n)ϱ2 r6 + O(r8 ) within the first order approximation in f .

3. Infinite cluster (n)

The following natural question can be stated. Is the limit limn→∞ σe correctly defined? The answer is (n) (n) negative since e2 is conditionally convergent. It was shown in [11,12] that the limit limn→∞ e2 depends on the shape of exterior curve enclosing the inclusions. In order to answer the question the regular square array is considered for definiteness. Without loss of generality the linear geometrical scale can be normalized in such a way that for any n the value nπr2 be equal to the fraction of inclusions. It does not change the result since the value M(n) from (n) (5) is dimensionless. We have e22 = e22 and e222 = e32 for the regular square array. The sum e3 converges absolutely to zero for the regular square array [13], hence, it does not depend on the method of summation. Introduce for shortness the undetermined values X = eπ2 and Am (m = 1, 2, 3) in A3 (f ) = ∑ m m=1,2,3 Am f . The limit of (5) becomes ∑ (∞) 2 3 M3 = 1 − f ϱX + (f ϱX) − (f ϱX) − A3 (f ) =: 1 + bm f m , (9) m=1,2,3 (∞)

(∞)

where M3 denotes the third order approximation for M(∞) = M3 be explained below. The limit formula

+ O(f 4 ). The correction A3 (f ) will

(∞)

σe =

1 + f ϱM3

(∞)

1 − f ϱM3

+ O(f 5 )

(10)

following from (8) can be asymptotically transformed into equation σe = 1 + 2ϱf + 2f 2 (b1 ϱ + ϱ2 ) + 2f 3 (b2 ϱ + 2b1 ϱ2 + ϱ3 ) 4

+ 2f (b3 ϱ +

b21 ϱ2

2

3

4

5

+ 2b2 ϱ + 3b1 ϱ + ϱ ) + O(f ).

(11)

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It was justified in [7,14] that the effective conductivity of the square array has the form σe = 1 + 2ϱf + 2ϱ2 f 2 + 2ϱ3 f 3 + 2ϱ4 f 4 + O(f 5 ).

(12)

(∞)

Comparison of (11) and (12) yields bm = 0, hence M3 = 1. The simplest formal way to satisfy (12) seems to take A3 (f ) = 0 and X = e2 = 0 by definition. However, this drastically complicates the limit n → ∞ in the local field (4). It seems natural to use the Eisenstein summation [8] e ∑ p,q

:=

lim

lim

M2 →+∞ M1 →+∞

M2 ∑

M1 ∑

.

(13)

q=−M2 p=−M1

Rayleigh [7] applied the Eisenstein summation to the problems in the periodic statement and established the (n) famous formula for the lattice sum S2 = π. It follows from [4] that e2 = limn→∞ e2 = π for macroscopically ∑ m isotropic composites. Then, the value A3 (f ) = m=1,2,3 (−f ϱ) must be introduced. 4. Discussion and conclusion We demonstrate in Section 2 that the local field in a finite cluster can give a formula for the effective conductivity valid only for clusters diluted in composite. This result has cast doubts on not justified declarations of other works on extension of Maxwell’s formalism for finite clusters to high concentrations. Study of infinite clusters in Section 3 leads to conditionally convergent series. Its interpretation can be (n) found in [11, Sec.2.4] and [12]. The limit e2 = limn→∞ e2 depends on the shape of exterior curve γ enclosing the inclusions. The term A3 (f ) is related to the total charge density over γ. The Eisenstein summation (13) transforms the terms of (4) into the Eisenstein functions for a periodic structure [13] and yields e2 = π. In the same time, Maxwell’s formalism is based on the dipole at infinity, i.e. the total charge density over γ. The Eisenstein limit n → ∞ presupposes the extra conditionally convergent part of the dipole moment A(f ) which has to be subtracted as it is made in (9) in the third order approximation in f . Thus, extension of Maxwell’s formalism to high concentrations requires a subtle mathematical treatment of the conditionally convergent series. It turns out that the analytical formulae from [15] for the effective constants of 2D elastic composites are actually based on Maxwell’s type formalism. This will be demonstrated analogously to the comparison of (11) and (12) in a separate paper. This implies that the final formulae (1) of [15] must be corrected, for instance, by taking e2 = e3 = 0 for the hexagonal array. This remark shows a computationally simple implementation of Maxwell’s formalism despite its subtle mathematical justification. Acknowledgment The paper was partially supported by National Science Centre, Poland, Research Project No. 2016/21/B/ST8/01181. References

[1] [2] [3] [4] [5] [6] [7]

J.C. Maxwell, Electricity and Magnetism, Vol. 1, Clarendon Press, 1873. T.C. Choy, Effective Medium Theory, Clarendon Press, 1999. O. Levy, E. Cherkaev, J. Appl. Phys. 114 (2013) 164102. V. Mityushev, N. Rylko, Quart. J. Mech. Appl. Math. 66 (2013) 241. G.M. Whitesides, B. Grzybowski, Science 295 (2002) 2418. R. Czapla, V. Mityushev, Math. Biosci. Eng. 14 (2017) 277. Rayleigh, Phil. Mag. 34 (1892) 481.

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[8] A. Weil, Elliptic Functions According To Eisenstein and Kronecker, Springer-Verlag, 1999. [9] V. Mityushev, Arch. Mech. 45 (1993) 211. [10] A.B. Movchan, N.V. Movchan, C.G. Poulton, Asymptotic Models of Fields in Dilute and Densely Packed Composites, World Scientific, 2002. [11] R.C. McPhedran, D.R. McKenzie, Proc. R. Soc. Lond. Ser. A 359 (1978) 45. [12] V. Mityushev, Arch. Mech. 49 (1997) 345. [13] P. Dryga´s, J. Math. Anal. Appl. 444 (2016) 1321. [14] W.T. Perrins, D.R. McKenzie, R.C. McPhedran, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 369 (1979) 207. [15] P. Dryga´s, V. Mityushev, Internat. J. Solids Struct. 97–98 (2016) 543.