Efficient computation of basic sums for random polydispersed composites

Abstract

The main goal of the paper was to develop algorithms and methods for computation of basic sums, the mathematical objects of great importance in computational materials science having applications to description of the representative volume element (RVE) and to the effective properties of 2D composites. The previously used algorithm had the exponential complexity. We propose a linearly complex algorithm. All the presented algorithms can be easily implemented in modern scientific computing packages, while maintaining both efficient calculations and a high level of abstraction. The proposed approach is applied to derivation of a polynomial approximation of the effective conductivity formula for 2D random material with non-overlapping circular inclusions with normally distributed radii. The obtained formulas are applied to the optimal packing problem of disks on the plane.

Appendix: Eisenstein–Rayleigh lattice sums

Consider the lattice $\mathcal {Q}$ (see Sect. 2). For definiteness, it is assumed that $\text {Im}\;\tau >0$, where $\tau =\omega _{2}/\omega _{1}$. The Eisenstein summation is defined by the iterative sum

$$\begin{aligned} \sum _{m_{1},m_{2}}=\lim _{N\rightarrow \infty }\sum _{m_{2}=-N}^{N}\left( \lim _{M\rightarrow \infty }\sum _{m_{1}=-M}^{M}\right) . \end{aligned}$$

(20)

The lattice sums are introduced as follows

$$\begin{aligned} S_{n}{:=}\sum _{m_{1},m_{2}}\;^{\prime }(m_{1}\omega _{1}+m_{2}\omega _{2})^{-n} \quad (n=2,3,\ldots ), \end{aligned}$$

(21)

where the prime means that $m_{1}$ and $m_{2}$ run over all integer numbers as in (20) except the pair $(m_{1},m_{2})=(0,0)$. The sum $S_{2}$ is conditionally convergent and understood in the sense of the Eisenstein summation (20). Though the rest sums (21) converge absolutely, the direct computations by (21) are problematic because of their slow convergence. The sum $S_{2}$ can be computed by a quick formula (Rylko 2000):

$$\begin{aligned} S_{2}= \left( \frac{\pi }{\omega _{1}}\right) ^{2}\left( \frac{1}{3}-8\sum _{m=1}^{\infty }\frac{mq^{2m}}{1-q^{2m}}\right) , \text { where }q=\exp \left( \pi i\tau \right) . \end{aligned}$$

(22)

It is known that $S_{n}=0$ for an odd n. For an even n, the sums (21) can be easily computed through the rapidly convergent infinite sums (Mityushev et al. 2008)

$$\begin{aligned} S_{4}= & {} \frac{1}{60}\left( \frac{\pi }{\omega _{1}}\right) ^{4}\left( \frac{4}{3}+320\sum _{m=1}^{\infty }\frac{m^{3}q^{2m}}{1-q^{2m}} \right) ,\; \end{aligned}$$

(23)

$$\begin{aligned} S_{6}= & {} \frac{1}{140} \left( \frac{\pi }{\omega _{1}}\right) ^{6}\left( \frac{8}{27}-\frac{448}{3}\sum _{m=1}^{\infty }\frac{m^{5}q^{2m}}{ 1-q^{2m}}\right) . \end{aligned}$$

(24)

The sums $S_{2n}$ ($n\ge 4$) are calculated by the recurrence formula (Mityushev et al. 2008)

$$\begin{aligned} S_{2n}=\frac{3}{\left( 2n+1\right) \left( 2n-1\right) \left( n-3\right) }\sum _{m=2}^{n-2}\left( 2m-1\right) \left( 2n-2m-1\right) S_{2m}S_{2(n-m)}. \end{aligned}$$

(25)

The Eisenstein series are defined as follows (Weil 1976)

$$\begin{aligned} E_{n}(z){:=}\sum _{m_{1},m_{2}}(z-m_{1}\omega _{1}-m_{2}\omega _{2})^{-n}\,,\;n=2,3,\ldots . \end{aligned}$$

(26)

Each of the functions (26) is doubly periodic and has a pole of order n at $z=0$. Further, it is convenient to define the value of $E_{n}(z)$ at the origin as $E_{n}(0){:=}S_{n}.$