# Efficient computation of basic sums for random polydispersed composites

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## 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.

## Keywords

Algorithms Basic sums Effective conductivity of composites Discrete multidimensional convolutions of functions Optimal packing problem## Mathematics Subject Classification

68Q25 68W05 74A40## Notes

### Acknowledgements

This material is based upon work supported by the National Centre for Research and Development under Grant No. NCN 2016/21/B/ST8/01181.

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