Continuous and discontinuous finite element methods for a peridynamics model of mechanics

In contrast to classical partial differential equation models, the recently developed peridynamic nonlocal continuum model for solid mechanics is an integro-differential equation that does not involve spatial derivatives of the displacement field. As a result, the peridynamic model admits solutions having jump discontinuities so that it has been successfully applied to fracture problems. The peridynamic model features a horizon which is a length scale that determines the extent of the nonlocal interactions. Based on a variational formulation, continuous and discontinuous Galerkin finite element methods are developed for the peridynamic model. Discontinuous discretizations are conforming for the model without the need to account for fluxes across element edges. Through a series of simple, one-dimensional computational experiments, we investigate the convergence behavior of the finite element approximations and compare the results with theoretical estimates. One issue addressed is the effect of the relative sizes of the horizon and the grid. For problems with smooth solutions, we find that continuous and discontinuous piecewise-linear approximations result in the same accuracy as that obtained by continuous piecewise-linear approximations for classical models. Piecewise-constant approximations are less robust and require the grid size to be small with respect to the horizon. We then study problems having solutions containing jump discontinuities for which we find that continuous approximations are not appropriate whereas discontinuous approximations can result in the same convergence behavior as that seen for smooth solutions. In case a grid point is placed at the locations of the jump discontinuities, such results are directly obtained. In the general case, we show that such results can be obtained through a simple, automated, abrupt, local refinement of elements containing the discontinuity. In order to reduce the number of degrees of freedom while preserving accuracy, we also briefly consider a hybrid discretization which combines continuous discretizations in regions where the solution is smooth with discontinuous discretizations in small regions surrounding the jump discontinuities.