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 $L^2$ norm at the rate $C_1 \Delta t + C_2 h^2/\epsilon^2$. Here, $\Delta t$, $h$, and $\epsilon$ are time step size, mesh size, and the size of the horizon or nonlocal length scale, respectively. Constants $C_1$ and $C_2$ are independent of $h$ and $\Delta t$ and depend on norms of the solution and nonlocal length scale. Several numerical examples involving pre-crack, void, and notch are considered, and the efficacy of the proposed nodal finite element discretization is analyzed.

翻译：本研究探讨近场动力学的节点有限元近似方法，其中节点位移满足每个网格节点处的近场动力学方程。针对非线性键基近场动力学模型，本文证明在精确解满足适当假设的条件下，采用时间中心差分与节点有限元离散化所得的离散解在 $L^2$ 范数下以 $C_1 \Delta t + C_2 h^2/\epsilon^2$ 的速率收敛于精确解。此处 $\Delta t$、$h$ 和 $\epsilon$ 分别表示时间步长、网格尺寸以及近场范围（非局部长度尺度）的尺寸。常数 $C_1$ 和 $C_2$ 与 $h$ 和 $\Delta t$ 无关，其取值取决于解的各阶范数及非局部长度尺度。研究通过含预置裂纹、孔洞和缺口的多组数值算例，验证了所提节点有限元离散化方法的有效性。