Schwarz’s alternating method in a matrix form and its applications to composites

摘要

Two-phase composites with n equal non-overlapping inclusions randomly embedded in the matrix are investigated. It is considered the case when the inclusions are bounded by some Lyapunov’s boundary curve. The problem is reduced to a vector-matrix problem of dimension n for one inclusion. The generalized alternating method of Schwarz applied to the vector-matrix problem is decomposed onto n scalar problems for one inclusion which are solved numerically by the method of integral equations for any smooth shape of the inclusions. A symbolic computation method is developed to solve the same problem by means of conformal mapping and functional equations. As a purpose, the effective conductivity of such models is exactly expressed through all geometrical and mechanical properties of its components.