A boundary method for the dynamic response of discrete lattices

Abstract

A novel boundary formulation is presented by applying the Boundary Element Method (BEM) to a dynamically loaded medium modelled as a discrete system. The two-dimensional medium is divided into a nonlinear discrete lattice in the near field, and a corresponding linear viscoelastic far field. The resulting boundary formulation is derived from the dynamic reciprocal work theorem and describes the far-field response through a Laplace domain force–displacement relation. The involved dynamic compliance matrix is composed of newly derived expressions for the Green’s functions of a viscoelastic half-plane of particles. It is demonstrated that the presented method yields a perfectly non-reflective boundary in the Laplace domain, without the need for artificial absorbing boundaries. Additionally, this contribution shows the successful time-domain application of the boundary method to a medium that exhibits non-smooth behaviour in the vicinity of a load source. In the time domain, the boundary equations are obtained by numerical application of the inverse Laplace transform, and the non-reflectiveness of the boundary is sensitive to the size of the time step. The presented method provides a consistent boundary approach for discrete lattices, and provides an alternative to continuum-based boundary methods for the dynamic response of solid media.