The Multiscale Analysis of Multiple Interacting Inclusions Problem: Finite Number of Interacting Inclusions

Abstract

A hybrid method based on the combination of the volume integral equation (VIE) method and the boundary integral equation (BIE) method is proposed for the micro-macro solution of elastostatic 2D and 3D multiscale problems in bounded or unbounded solids containing interacting multiple inclusions of essentially different scale. The hybrid micro-macro formulation allows decomposition of the complete problem into two associated subproblems, one residing entirely at the micro-level and the other at the macro-level at each iteration. The efficiency of the standard iterative scheme of the BIE and VIE methods for the singular integral equations involved is enhanced by the use of a modification in the spirit of a subtraction technique as well as by the advantageous choice of the initial analytical approximation for interacting inclusions (micro-level) in an unbounded medium subjected to inhomogeneous loading. The latter is evaluated by the macro-scale BIE technique capable of handling complex finite geometries and mixed boundary conditions. The iteration method proposed converges rapidly in a wide class of problems considered with high matrix-inclusion elastic contrast, with continuously varying anisotropic and nonlinear elastic properties of inclusions, as well as with sizes of interacting inclusions differing by a factor varying in the interval from 1 to 107. The accuracy and efficiency of the method are examined through comparison with results obtained from finite-element analysis and boundary element analysis as well as from analytical solution.