A Fast Collocation Method with Inscribed Polygonal Approximation of the Peridynamic Neighborhood for Bond-Based Linear Viscoelastic Peridynamic Models

The peridynamic (PD) theory is a reformulation of the classical theory of continuum solid mechanics, which yields an integro-differential equation that does not involve spatial derivatives of the displacement field. Therefore, it is particularly suitable for the description of cracks and their evolution in materials. Due to the nonlocal property of PD models, classical numerical methods usually generate dense stiffness matrices which require $\mathscr{O}(N^3)$ computational work and $\mathscr{O}(N^2)$ memory, where $N$ is the number of spatial unknowns. In this paper, we develop a fast collocation method for a bond-based linear viscoelastic peridynamic model in two space dimensions by exploring the structure of the stiffness matrix. The method has a computational work count of $\mathscr{O}(N {\rm log}N)$ per Krylov subspace iteration and $\mathscr{O}(N)$ memory requirements. In our model, the peridynamic neighborhood is a two-dimensional circular domain, so that one has to deal with the intersection of such circles and rectangular meshes used in the collocation method. We employ an inscribed polygon to approximate the peridynamic neighborhood. This improves the numerical accuracy of solution and reduces programming complexity. Numerical results show the utility of the proposed method.