An hp-Galerkin method with fast solution for linear peridynamic models in one dimension

Abstract

The computational work and memory requirement are bottlenecks for Galerkin finite element methods for peridynamic models because of their non-locality. In this paper, fast Galerkin and hp-Galerkin finite element methods are introduced and analyzed to solve a steady-state peridynamic model. We present a fast solution technique to accelerate non-square Toeplitz matrix-vector multiplications arising from piecewise-linear, piecewise-quadratic and piecewise-cubic Galerkin methods. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from O(N 3) required by traditional methods to O(Nlog 2 N) and the memory requirement from O(N 2) to O(N) without using any lossy compression, where N is the number of unknowns. The peridynamic model admits solutions having jump discontinuities. For problems with discontinuous solutions, we therefore introduce a piecewise-constant Galerkin method and give an hand p-refinement algorithm. Then, we develop fast hp-Galerkin methods based on hybrid piecewise-constant/piecewise-linear and piecewiseconstant/piecewise-quadratic finite element approximations. The new method reduces the computational work from O(N 3) required by the traditional methods to O(Nlog 2 N) and the memory requirement from O(N 2) to O(N).

Peridynamics review

“…In order to reduce the computational cost in PD computations, Wang and Tian [133] proposed a fast Galerkin method with efficient matrix assembly and storage for a PD model. Later, their proposed model was extended by Liu et al [134] to higherorder discretizations and hp adaptivity. Interesting studies were presented by Du et al [135,136] in which an adaptive FEM was developed for non-local diffusion equations and PD models by establishing a posteriori error analysis.…”

Bond-based peridynamics, a survey prospecting nonlocal theories of fluid-dynamics

“…Application of continuous and discontinuous Galerkin finite element methods to PD has been explored in [62] and validated against 1D peridynamics exact solutions. In [63] a fast and cost-efficient Galerkin method developed in [64] has been extended and improved for 1D linearized PD static problems providing hp-adaptivity, reducing computational efforts and memory usage. In [65] the implementation of a coupled PD-FEM approach in commercial software ABAQUS® has been performed, employing mesh coupling techniques developed in [66][67][68], with good results, whereas, recently, in [69] a PD-FEM coupled approach has been carried out with the commercial software ANSYS® for fatigue prediction in materials.…”