Connections between peridynamics and graph Laplacian for diffusion problems

Abstract

Formulations for diffusion processes based on graph Laplacian kernels have been recently used to solve linear transient heat transfer problems with insulated boundary conditions by way of a spectral-based semi-analytical approach. This has been called the “spectral graph” (SG) approach. In this paper we show that the meshfree discretizations for corresponding peridynamic models have a similar graph structure and lead to the same general equation as the SG approach. In this sense, the SG approach can be seen as a particular case of a peridynamic formulation. For the transient heat diffusion, we explain some differences between the SG approach and peridynamics, related to calibration and discretization procedures. We use a 1D heat diffusion example to highlight some limitations the spectral-based semi-analytical method in the SG approach has compared with the direct time-integration normally used in computing solutions to peridynamic models. We also introduce an extension of the semi-analytical approach to diffusion problems with Dirichlet boundary conditions.