Enforcing local boundary conditions in peridynamic models of diffusion with singularities and on arbitrary domains

Abstract

Imposing local boundary conditions and mitigating the surface effect at free surfaces in peridynamic (PD) models are often desired. The fictitious nodes method (FNM) “extends” the domain with a thin fictitious layer of thickness equal to the PD horizon size, and is a commonly used technique for these purposes. The FNM, however, is limited, in general, to domains with simple geometries. Here we introduce an algorithm for the mirror-based FNM that can be applied to arbitrary domain geometries. The algorithm automatically determines mirror nodes (in the given domain) of all fictitious nodes based on approximating, at each fictitious node, the “generalized” (or nonlocal) normal vector to the domain boundary. We tested the new algorithm for a peridynamic model of a classical diffusion problem with a flux singularity on the boundary. We show that other types of FNMs exhibit “pollution” of the solution far from the singularity point, while the mirror-based FNM does not and, in addition, shows a significantly faster rate of convergence to the classical solution in the limit of the horizon going to zero. The new algorithm is then used for mirror-based FNM solutions of diffusion problems in domains with curvilinear boundaries and with intersecting cracks. The proposed algorithm significantly improves the accuracy near boundaries of domains of arbitrary shapes, including those with corners, notches, and crack tips.

Graphical Abstract