Abstract
A fast and accurate algorithm for the computation of effective electric and mathematically equivalent properties of composites with nonsmooth interfaces is reported. The algorithm is based on an integral equation reformulation of the electrostatic partial differential equation and a fast hierarchical technique for potential field evaluation. In a numerical example, 200 large and strongly inhomogeneous aggregates of randomly overlapping disks are solved with a relative error of 0.0005.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
B. I. Halperin, S. Feng, and P. N. Sen, Differences between lattice and continuum percolation transport exponents, Phys. Rev. Lett. 54:2391 (1985).
J. P. Clerc, G. Giraud, J. M. Laugier, and J. M. Luck, The electrical conductivity of binary disordered systems, percolation clusters, fractals and related models, Advances in Physics 39:191 (1990).
L. M. Schwartz, J. R. Banavar, and B. I. Halperin, Biased-diffusion calculations of electrical transport in inhomogeneous continuum systems, Phys. Rev. B 40:9155 (1989).
J. Tobochnik, D. Laing, and G. Wilson, Random-walk calculation of conductivity in continuum percolation, Phys. Rev. A 41:3052 (1990).
M. M. Tomadakis and S. V. Sotirchos, Transport through random arrays of conductive cylinders dispersed in a conductive matrix, J. Chem. Phys. 104:6893 (1996).
E. J. Garboczi, M. F. Thorpe, M. S. DeVries, and A. R. Day, Universal conductivity curve for a plane containing random holes, Phys. Rev. A 43:6473 (1991).
T. Elam, A. R. Kerstein, and J. J. Rehr, Critical properties of the void percolation problem for spheres, Phys. Rev. Lett. 52:1516 (1984).
B. Berkowitz and R. Knight, Continuum percolation conductivity exponents in restricted domains, J. Stat. Phys. 80:1415 (1995).
G. W. Milton, Bounds on the electromagnetic, elastic, and other properties of two-component composites, Phys. Rev. Lett. 46:542 (1981).
G. W. Milton, Bounds on the elastic and transport properties of two-component composites, J. Mech. Phys. Solids 30:177 (1982).
S. Torquato and J. D. Beasley, Bounds on the effective thermal conductivity of dispersions of fully penetrable cylinders, Int. J. Engng. Sci. 24:415 (1986).
C. J. Lobb and M. G. Forrester, Measurement of nonuniversal critical behavior in a two-dimensional continuum percolating system, Phys. Rev. B 35:1899 (1990).
K. H. Han, Z. S. Lim, and S. I. Lee, Experimental study of the conductivity exponent in a two-dimensional Swiss cheese percolating network, Physica B 167:185 (1990).
V. Rokhlin, Rapid solution of integral equations of classical potential theory, J. Comput. Phys. 60:187-207 (1985).
L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys. 73:325 (1987).
J. Carrier, L. Greengard and V. Rokhlin, A fast adaptive multipole algorithm for particle simulations, SIAM J. Sci. and Stat. Comput. 9:669-686 (1988).
J. Helsing, Thin bridges in isotropic electrostatics, J. Comp. Phys. 127:142 (1996).
L. Greengard and M. Moura, On the numerical evaluation of electrostatic fields in composite materials, Acta Numerica 1994(Cambridge University Press, Cambridge, 1994), pp. 379-410.
H. Cheng and L. Greengard, On the numerical evaluation of electrostatic fields in dense random dispersion of cylinders, J. Comput. Phys. 136:629 (1997).
Y. Saad and M. H. Schultz, GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7:856-869 (1986).
Z. Hashin and S. Shtrikman, A variational approach to the theory of effective magnetic permeability of multiphase materials, J. Appl. Phys. 33:3125 (1962).
D. A. G. Bruggeman, Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen, Ann. Physik 24:636 (1935).
L. Greengard and V. Rokhlin, A new version of the Fast Multipole Method for the Laplace equation in three dimensions, Acta Numerica 1997(Cambridge University Press, Cambridge, 1997), pp. 229-269.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Helsing, J. A High-Order Accurate Algorithm for Electrostatics of Overlapping Disks. Journal of Statistical Physics 90, 1461–1473 (1998). https://doi.org/10.1023/A:1023204117016
Issue Date:
DOI: https://doi.org/10.1023/A:1023204117016