Efficient Solutions for Nonlocal Diffusion Problems Via Boundary-Adapted Spectral Methods

Abstract

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 ( log ). 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.

Crack nucleation in brittle and quasi-brittle materials: A peridynamic analysis

“…In principle, Eq. (1) can be discretized using the finite element method (FEM) [16,38], meshfree direct discretization [9], a combination of both [16,39,40], pseudo-spectral methods [41], or any other method suitable for numerically computing the solution to an integro-differential equation (or an integral equation for the static case). Here we use the meshfree discretization, which makes it easiest to handle damage and fracture [42,43].…”

A stochastic multiscale peridynamic model for corrosion-induced fracture in reinforced concrete

“…(1) can be solved by any method that can solve integro-differential equations, including mesh-free direct discretization [47], the finite element method (FEM) [62,63], or a combination of both in which the FEM is used far from cracks, and the meshfree discretization is used where damage happens [63][64][65]. Spectral methods can be alternative approaches to achieve efficient peridynamic computations [66]. Here we use the meshfree discretization, which makes it easiest to handle damage and fracture [31,67].…”

Dirichlet absorbing boundary conditions for classical and peridynamic diffusion-type models

“…The mathematical description of many phenomena from areas such as fluid dynamics, chemistry, biology, information, environmental and materials sciences is governed by diffusion-type equations. In this regard, numerous numerical methods (based on the classical local diffusion) have so far been employed, for example the finite element method (FEM), the finite difference method (FDM), the boundary element method (BEM), and meshfree methods (see, e.g., [4,6,25,31]). At the macroscale, most diffusion processes can be described well by local models based on Fourier's law (heat conduction) as well as Fick's law (mass transport).…”