A fast collocation method for a static bond-based linear peridynamic model

Abstract

We develop a fast collocation method for a steady-state bond-based linear peridynamic model in two space dimensions. The method reduces the computational work from O(N2) per Krylov subspace iteration in a traditional collocation method to O(NlogN) and the memory requirement from O(N2) in a collocation method to O(N), where N is the number of unknowns in the discrete system. Furthermore, the method reduces the computational work of evaluating and assembling the stiffness matrix, which often constitutes a major portion of the overall computational work, from O(N2) in a collocation method to O(N). The significant reduction of the CPU times and storage in the fast collocation method is achieved by carefully exploring the structure of the stiffness matrix of the collocation method without any lossy compression involved. In other words, the fast method is evaluated in an equivalent but more efficient manner. Therefore, the fast method generates identical numerical solutions as the traditional collocation method does and naturally inherits the stability and convergence properties that were already proved for a traditional collocation method. Numerical results are presented to show the utility of the fast method.