We consider the interface boundary value problem which arises in the evaluation of electrostatic fields in composite materials con-ena .
sisting of dense random dispersions of cylinders in a uniform back-A variety of numerical methods can be used for direct ground. This is a well-studied problem from the viewpoint of hosolution of the field equation, including finite difference mogenization and effective medium theory, but one for which and finite element methods, but we will restrict our attenaccurate numerical simulations have been difficult to obtain. Size tion here to methods based on potential theory. The first effects, in particular, have been neglected due to the expense of solving the field equation in the presence of large numbers of close-such method is due to Lord Rayleigh, who considered the to-touching inclusions. Such features require very fine discretizproblem of conductivity through a regular array of disks ations, even with the use of adaptive gridding, and cause the linear or spheres . More recent work using this approach systems which arise to be highly ill-conditioned. In this paper, we includes 11, 30, 35,. More general present a new integral equation method for the solution of the integral equation formulations of the interface problem interface problem which uses a recently developed method of images to resolve the close-to-touching interactions and the fast can be used to treat composites with inclusions multipole method to compute far field interactions. Only minutes of arbitrary shape [11, 17], and even anisotropic material of workstation time are needed to solve the field equation with properties [15]. While these integral equations result in thousands of inclusions, allowing us to carry out large-scale statistidense N ฯซ N linear systems, where N is the number of cal studies of the effective conductivity of random two-phase materipoints in the discretization of the interface, fast multipoleals at a variety of volume fractions and contrast ratios. แฎ 1997 Academic Press accelerated iterative schemes require only O(N) work for the solution process [11, 15, 37]. Despite this methodological advance, however, certain problems have been out of by the Applied Mathematical Sciences Program of the U.S. Department of method (FMM) [9, 12, 37], which allows us to compute Energy under Contract DEFG02-88ER25053 and by a NSF Presidential Young Investigator Award.
far field interactions efficiently. The resulting scheme can 629