Efficient solutions for nonlocal diffusion problems via boundary-adapted spectral methods
This work provides an efficient numerical method for nonlocal diffusion problems, which is important for computational mechanics but represents an incremental improvement over existing spectral methods.
The paper introduces a boundary-adapted spectral method for peridynamic diffusion problems that achieves O(N log N) scaling by transforming convolution integrals into Fourier multiplications, using volume penalization to handle arbitrary boundary conditions. Convergence studies demonstrate high accuracy for Dirichlet and Neumann conditions.
We introduce an efficient boundary-adapted spectral method for peridynamic diffusion problems with arbitrary boundary conditions. The spectral approach transforms the convolution integral in the peridynamic formulation into a multiplication in the Fourier space, resulting in computations that scale as O(NlogN). The limitation of regular spectral methods to periodic problems is eliminated using the volume penalization method. We show that arbitrary boundary conditions or volume constraints can be enforced in this way to achieve high levels of accuracy. To test the performance of our approach we compare the computational results with analytical solutions of the nonlocal problem. The performance is tested with convergence studies in terms of nodal discretization and the size of the penalization parameter in problems with Dirichlet and Neumann boundary conditions.