Fast and accurate calculations of structural parameters for suspensions

The calculation of effective properties of periodic suspensions is often problematic.Particularly difficult are calculations involving unit cells with many, close to touching, inclusions and high desired accuracy. In this paper we apply the conjugate gradient method and the fast multipole method to simplify calculations of this kind. We show how to dramatically speed up the computation of the effective conductivities and structural parameters for suspensions of disks and spheres. This enables accurate treatment of unit cells with thousands of inclusions. Direct estimates of the effective conductivity are compared with estimates via bounds. Accuracy of twelve digits is obtained for a suspension of disks which has been studied previously, but for which no accurate digit has been determined.

[1]  G. A. Baker.  Essentials of Padé approximants , 1975 .

[2]  Graeme W. Milton,et al.  Bounds on the transport and optical properties of a two‐component composite material , 1981 .

[3]  Ross C. McPhedran,et al.  Bounds and exact theories for the transport properties of inhomogeneous media , 1981 .

[4]  R. McPhedran,et al.  A comparison of two methods for deriving bounds on the effective conductivity of composites , 1982 .

[5]  A. Sangani,et al.  Bulk thermal conductivity of composites with spherical inclusions , 1988 .

[6]  Graeme W. Milton,et al.  Asymptotic studies of closely spaced, highly conducting cylinders , 1988, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[7]  O. Bruno.  The effective conductivity of an infinitely interchangeable mixture , 1990 .

[8]  M. Thorpe,et al.  The conductivity of a sheet containing inclusions with sharp corners , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[9]  David R. McKenzie,et al.  Transport properties of regular arrays of cylinders , 1979, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[10]  M. Beran.  Use of the vibrational approach to determine bounds for the effective permittivity in random media , 1965 .

[11]  Melvin N. Miller.  Bounds for Effective Electrical, Thermal, and Magnetic Properties of Heterogeneous Materials , 1969 .

[12]  V. Rokhlin,et al.  Rapid Evaluation of Potential Fields in Three Dimensions , 1988 .

[13]  Graeme W. Milton,et al.  Multicomponent composites, electrical networks and new types of continued fraction I , 1987 .

[14]  N. Phan-Thien,et al.  New bounds on the effective thermal conductivity of N-phase materials , 1982, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[15]  Leslie Greengard,et al.  A numerical study of the ζ2 parameter for random suspensions of disks , 1995 .

[16]  Johan Helsing,et al.  Bounds to the conductivity of some two‐component composites , 1993 .

[17]  David J. Bergman,et al.  The dielectric constant of a simple cubic array of identical spheres , 1979 .

[18]  S. Childress,et al.  Macroscopic Properties of Disordered Media , 1982 .

[19]  Salvatore Torquato,et al.  Bounds on the conductivity of a random array of cylinders , 1988, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[20]  L. Ungar,et al.  Application of the boundary element method to dense dispersions , 1988 .

[21]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[22]  D. R. McKenzie,et al.  The conductivity of lattices of spheres I. The simple cubic lattice , 1978, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[23]  A. Sangani,et al.  Transport Processes in Random Arrays of Cylinders. I. Thermal Conduction , 1988 .

[24]  L. Rayleigh,et al.  LVI. On the influence of obstacles arranged in rectangular order upon the properties of a medium , 1892 .

[25]  J. Brady,et al.  The effective conductivity of random suspensions of spherical particles , 1991, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[26]  Hinsen,et al.  Dielectric constant of a suspension of uniform spheres. , 1982, Physical review. B, Condensed matter.

[27]  D. J. Bergman,et al.  The optical properties of cermets from the theory of electrostatic resonances , 1982 .

[28]  Salvatore Torquato,et al.  Effective conductivity of hard-sphere dispersions , 1990 .

[29]  D. Mckenzie,et al.  Electrostatic and optical resonances of arrays of cylinders , 1980 .

[30]  S. Prager,et al.  Improved Variational Bounds on Some Bulk Properties of a Two‐Phase Random Medium , 1969 .

[31]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[32]  Graeme W. Milton,et al.  Bounds on the elastic and transport properties of two-component composites , 1982 .

[33]  L. Poladian,et al.  Effective transport properties of periodic composite materials , 1986, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[34]  David J. Bergman,et al.  The dielectric constant of a composite material—A problem in classical physics , 1978 .

[35]  S. Shtrikman,et al.  A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials , 1962 .