A Fast Implementation of the Linear Bond-Based Peridynamic Beam Model

Abstract

While the theory of peridynamics (PD) holds significant potential in engineering, its application is often limited by the significant computational costs by the nonlocality of PD. This research is based on a three-dimensional (3D) complex Timoshenko beam structure with six degrees of freedom. We propose a fast meshfree method based on the linear bond-based PD model of the stiffness matrix structure by ingeniously using the matrix decomposition strategy to maintain the Teoplitz structure of the stiffness matrix. This method significantly reduces the amount of calculation and storage without losing accuracy, reduces the amount of calculation from $\mathcal{O}(N^2)$ to $\mathcal{O}(N{\rm log}N),$ and decreases the storage capacity from $\mathcal{O}(N^2)$ to $\mathcal{O}(N).$ We validate the effectiveness of our approach through numerical examples, particularly in multi-beam structures. We demonstrate that our method realizes algorithm acceleration in numerical simulations of multi-beam structures subjected to static concentrated loads.