A Fast Solver of Collocation Method for a Three-Dimensional Bond-Based Linear Peridynamic Model

The peridynamic (PD) theory tends to provide a better description than the classical continuum mechanics theory in the context of long-range forces or discontinuities. However, due to the nonlocal properties of PD models, the numerical methods employed in these models yield stiffness matrices with very complicated structures, especially for three-dimensional problems. Fast algorithms usually require the computational domain to be rectangular in two space dimensions and cuboidal in three space dimensions to ensure that the resulting stiffness matrices have a Toeplitz-like structure. In this work, we construct an accurate and efficient solver of the collocation method for a three-dimensional bond-based linear PD model defined on a relatively general geometry. The discretization for PD models on cubic domains by the collocation method can lead to Block-Block-Toeplitz-Toeplitz-Block-Block-Toeplitz-Toeplitz-Block-Block (B-BTTB-BTTB-B) matrices. Based on this special structure, we optimize the computational work of the traditional solver from \(O(N^3)\) to \(O(N^2log N)\) and reduce the memory requirement of the traditional solver from \(O(N^2)\) to \(O(N)\), where \(N\) is the number of spatial unknowns. More importantly, the fast method remains applicable to non-cuboidal geometries via domain extension into matching cubic areas and then carefully handling the nodes in the extended domains. Numerical experiments are performed to show the accuracy and efficiency of the proposed method.