Circulant preconditioned iterative methods for peridynamic model simulation

The Galerkin finite element formulation is employed to discretize the peridynamic model for microelastic materials on a finite bar. The coefficient matrix of the resulting system possesses the symmetric Toeplitz-plus-tridiagonal structure. As the symmetric Toeplitz-plus-tridiagonal matrix–vector multiplication can be efficiently implemented via using the fast Fourier transforms (FFTs), which can reduce the computational cost of the matrix–vector product, in this paper we consider the conjugate gradient (CG) method with a circulant preconditioner to solve the discretized linear systems. Moreover, the spectrum of the preconditioned matrix is proven to be mostly clustered around 1. Thus the accelerating benefits of circulant preconditioners are theoretically guaranteed. Numerical experiments are reported to demonstrate the effectiveness and robustness of circulant preconditioners in terms of the number of iterations and CPU time. Finally, the comprehensive performances evaluation of different circulant preconditioners are also given.