Nodal finite element approximation of peridynamics

This work considers the nodal finite element approximation of peridynamics, in which the nodal displacements satisfy the peridynamics equation at each mesh node. For the nonlinear bond-based peridynamics model, it is shown that, under the suitable assumptions on an exact solution, the discretized solution associated with the central-in-time and nodal finite element discretization converges to the exact solution in L² norm at the rate C₁ Δ t + C₂ h²/ε². Here, Δ t, h, and ε are time step size, mesh size, and the size of the horizon or nonlocal length scale, respectively. Constants C₁ and C₂ are independent of h and Δ t and depend on the norms of the exact solution. Several numerical examples involving pre-crack, void, and notch are considered, and the efficacy of the proposed nodal finite element discretization is analyzed.