Peridynamics and nonlocal diffusion models: Fast numerical methods

Abstract

We outline the recent developments of fast numerical methods for linear nonlocal diffusion and peridynamic models in one and two space dimensions. We show how the analysis was carried out to take full advantage of the structure of the stiffness matrices of the numerical methods in its storage, evaluation, and assembly and in the efficient solution of the corresponding numerical schemes. This significantly reduces the computational complexity and storage of the numerical methods over conventional ones, without using any lossy compression. For instance, we would use the same numerical quadratures for conventional methods to evaluate the singular integrals in the stiffness matrices, except that we only need to evaluate O.(N) of them instead of O(N2) of them. Numerical results are presented to show the utility of these fast methods.