Selecting the kernel in a peridynamic formulation: A study for transient heat diffusion

The kernel in a peridynamic diffusion model represents the detailed interaction between points inside the nonlocal region around each material point. Several versions of the kernel function have been proposed. Although solutions associated with different kernels may all converge, under the appropriate discretization scheme, to the classical model when the horizon goes to zero, their convergence behavior varies. In this paper, we focus on the particular one-point Gauss quadrature method of spatial discretization of the peridynamic diffusion model and study the convergence properties of different kernels with respect to convergence to the classical, local, model for transient heat transfer equation in 1D, where exact representation of geometry is available. The one-point Gauss quadrature is the preferred method for discretizing peridynamic models because it leads to a meshfree model, well suited for problems with damage and fracture. We show the equivalency of two definitions for the peridynamic heat flux. We explain an apparent paradox and discuss a common pitfall in numerical approximations of nonlocal models and their convergence to local models. We also analyze the influence of two ways of imposing boundary conditions and that of the “skin effect” on the solution. We explain an interesting behavior of the peridynamic solutions for different horizon sizes, the crossing of m-convergence curves at the classical solution value that happens for one of the ways of implementing the classical boundary conditions. The results presented here provide practical guidance in selecting the appropriate peridynamic kernel that makes the one-point Gauss quadrature an “asymptotically compatible” scheme. These results are directly applicable to any diffusion-type model, including mass diffusion problems.