### Abstract

The effective conductivity of two-component composites can be tightly bounded through the knowledge of structural parameters. While the first- and second-order parameters are known analytically for isotropic materials, the third and higher order parameters are generally not. Their evaluation has, therefore, become the subject of much research. In particular, the third-order structural parameter ζ_{2} has been computed many times. Interface methods, beginning with Rayleigh, have proven successful for periodic composites with simple unit cells. Statistical methods, involving three-point correlation functions, work well for dilute random suspensions. Composites consisting of complicated, dense suspensions have been much more difficult to treat. In this article, we illustrate how one can greatly accelerate the computation of structural parameters with interface methods, so that these methods can be applied to dense suspensions with tens of thousands of randomly placed inclusions per unit cell. We implement a numerical scheme, based on the fast multipole method, for which the amount of work grows linearly with the number of inclusions per unit cell and quadratically with the logarithm of the desired precision. By incorporating a Monte Carlo sampling technique, we have computed values of ζ_{2} for the random suspension of disks at 20 volume fractions between 0.50 and 0.69. These tabulated values are accurate to at least three digits and improve on the best previous estimates by a factor of between 30 and 100.

Original language | English (US) |
---|---|

Pages (from-to) | 2015-2019 |

Number of pages | 5 |

Journal | Journal of Applied Physics |

Volume | 77 |

Issue number | 5 |

DOIs | |

State | Published - 1995 |

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### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

*Journal of Applied Physics*,

*77*(5), 2015-2019. https://doi.org/10.1063/1.358838

**A numerical study of the ζ2 parameter for random suspensions of disks.** / Greengard, Leslie; Helsing, Johan.

Research output: Contribution to journal › Article

*Journal of Applied Physics*, vol. 77, no. 5, pp. 2015-2019. https://doi.org/10.1063/1.358838

}

TY - JOUR

T1 - A numerical study of the ζ2 parameter for random suspensions of disks

AU - Greengard, Leslie

AU - Helsing, Johan

PY - 1995

Y1 - 1995

N2 - The effective conductivity of two-component composites can be tightly bounded through the knowledge of structural parameters. While the first- and second-order parameters are known analytically for isotropic materials, the third and higher order parameters are generally not. Their evaluation has, therefore, become the subject of much research. In particular, the third-order structural parameter ζ2 has been computed many times. Interface methods, beginning with Rayleigh, have proven successful for periodic composites with simple unit cells. Statistical methods, involving three-point correlation functions, work well for dilute random suspensions. Composites consisting of complicated, dense suspensions have been much more difficult to treat. In this article, we illustrate how one can greatly accelerate the computation of structural parameters with interface methods, so that these methods can be applied to dense suspensions with tens of thousands of randomly placed inclusions per unit cell. We implement a numerical scheme, based on the fast multipole method, for which the amount of work grows linearly with the number of inclusions per unit cell and quadratically with the logarithm of the desired precision. By incorporating a Monte Carlo sampling technique, we have computed values of ζ2 for the random suspension of disks at 20 volume fractions between 0.50 and 0.69. These tabulated values are accurate to at least three digits and improve on the best previous estimates by a factor of between 30 and 100.

AB - The effective conductivity of two-component composites can be tightly bounded through the knowledge of structural parameters. While the first- and second-order parameters are known analytically for isotropic materials, the third and higher order parameters are generally not. Their evaluation has, therefore, become the subject of much research. In particular, the third-order structural parameter ζ2 has been computed many times. Interface methods, beginning with Rayleigh, have proven successful for periodic composites with simple unit cells. Statistical methods, involving three-point correlation functions, work well for dilute random suspensions. Composites consisting of complicated, dense suspensions have been much more difficult to treat. In this article, we illustrate how one can greatly accelerate the computation of structural parameters with interface methods, so that these methods can be applied to dense suspensions with tens of thousands of randomly placed inclusions per unit cell. We implement a numerical scheme, based on the fast multipole method, for which the amount of work grows linearly with the number of inclusions per unit cell and quadratically with the logarithm of the desired precision. By incorporating a Monte Carlo sampling technique, we have computed values of ζ2 for the random suspension of disks at 20 volume fractions between 0.50 and 0.69. These tabulated values are accurate to at least three digits and improve on the best previous estimates by a factor of between 30 and 100.

UR - http://www.scopus.com/inward/record.url?scp=0010811879&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0010811879&partnerID=8YFLogxK

U2 - 10.1063/1.358838

DO - 10.1063/1.358838

M3 - Article

VL - 77

SP - 2015

EP - 2019

JO - Journal of Applied Physics

JF - Journal of Applied Physics

SN - 0021-8979

IS - 5

ER -