A fast convolution-based method for peridynamic transient diffusion in arbitrary domains

Abstract

We introduce a fast convolution-based method (FCBM) for solving linear and a certain class of nonlinear peridynamic (PD) transient diffusion problems in 1D, 2D, and 3D. The method exploits the convolutional structure of the PD diffusion operator to compute it efficiently by using the fast Fourier transform (FFT). A new “embedded constraint” (EC) strategy allows using the Fourier transform on irregular domains and imposing arbitrary nonlocal boundary conditions. The complexity of the new method is O(Nlog2N), compared with O(N2) for the conventional peridynamic meshfree or finite element solvers of the same problem. We find quadratic convergence rates in terms of spatial discretization to the exact nonlocal solutions in 1D and 2D problems. Numerical tests show substantial efficiency gains when using the FCBM-EC method compared to the meshfree discretization method for problems with a larger number of nodes inside the nonlocal region. Further speedup is achieved with Matlab’s intrinsic FFT functions for computations on GPUs or multiple CPUs. An example in 3D with over 1 billion degrees of freedom over tens of thousands of time-steps, is solved by the new method on a single CPU in a matter of days. The same problem would have required over a century to complete with the commonly used meshfree discretization.