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 a solution in the L2 norm at the rate C1Δt+C2h2/ϵ2. Here, Δt, 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 Δt 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.

中文翻译：

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近场动力学的节点有限元近似


这项工作考虑了近场动力学的节点有限元近似，其中节点位移满足每个网格节点的近场动力学方程。对于基于非线性键的近场动力学模型，结果表明，在对精确解的适当假设下，与中心时间和节点有限元离散化相关的离散解以 C1Δt+C2h2/ε2 的速率收敛到 L2 范数中的解。其中，Δt、h 和 ε 分别是时间步长、网格大小以及水平或非局部长度尺度的大小。常数 C1 和 C2 与 h 和 Δt 无关，取决于解的范数和非局部长度尺度。考虑了几个涉及预裂纹、空隙和缺口的数值示例，并分析了所提出的节点有限元离散化的有效性。