The complex variable fast multipole boundary element method for the analysis of strongly inhomogeneous media

Abstract

The paper aims to develop the efficient method tailored for accurate, robust and stable calculations of 2D local fields in strongly inhomogeneous materials with arbitrary interaction conditions on multiple contacts of structural blocks. The method is also to be of immediate use for solving homogenisation problems. To reach the goal we employ: (i) special forms of the complex variable singular and hypersingular integral equations with the densities representing those physical quantities, which enter the contact conditions; (ii) circular-arc boundary elements (in addition to straight elements) for smooth approximation of smooth parts of the external boundaries and contacts; (iii) higher order approximations of densities, which account for arbitrary power asymptotics of physical fields near singular points (crack tips, corner points, common apexes of structural blocks); (iv) analytical recurrent evaluation of all influence coefficients; (v) analytical recurrent evaluation of all moments; (vi) the complex variable fast multipole method (CV FMM), for solving the resulting system of the complex variable boundary element method (CV BEM), with large number (up to million) of unknowns. As a result, we obtain free of numerical integration, higher-order CV fast multipole boundary element method (CV FM-BEM) for a medium with multiple structural elements and multiple singular points. In the due course, we suggest the simplified starting quadrature formulae for singular boundary elements, the adjustment of the procedure for building the hierarchical tree and the proper choice of the key parameters of the developed CV FM-BEM: the number of elements in a leaf; the number of moments in the truncated Taylor expansions; the reasonable tolerance, when iteratively solving the system by the FMM. Numerical examples illustrate the abilities of the method developed, as regard to local fields in strongly inhomogeneous structures with multiple singular points. The study of local fields shows application of the method to finding extreme distributions of stress intensity factors in a medium with many cracks, which may intersect. The homogenisation problem is solved, as well.