In this work we estimate the convergence rate for time stepping schemes applied to nonlocal dynamic fracture modeling. Here we use the nonlocal formulation given by the bond based peridynamic equation of motion. We begin by establishing the existence of $ H^2 $ peridynamic solutions over any finite time interval. For this model the gradients can become large and steep slopes appear and localize when the non-locality of the model tends to zero. In this treatment spatial approximation by finite elements are used. We consider the central-difference scheme for time discretization and linear finite elements for discretization in the spatial variable. The fully discrete scheme is shown to converge to the actual $ H^2 $ solution in the mean square norm at the rate $ C_t\Delta t +C_s h^2/\epsilon^2 $. Here $ h $ is the mesh size, $ \epsilon $ is the length scale of nonlocal interaction and $ \Delta t $ is the time step. The constants $ C_t $ and $ C_s $ are independent of $ \Delta t $, and $ h $. In the absence of nonlinearity a CFL like condition for the energy stability of the central difference time discretization scheme is developed. As an example we consider Plexiglass and compute constants in the a-priori error bound.

中文翻译：

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非局部动态裂缝模型的有限元逼近

在这项工作中，我们估计了应用于非局部动态裂缝建模的时间步长方案的收敛速度。在这里，我们使用基于键的运动绕动力学方程式给出的非局部公式。我们首先确定在任何有限的时间间隔内存在$ H ^ 2 $周边动力学解。对于此模型，当模型的非局部性趋于零时，坡度可能会变大，并且会出现陡峭的坡度并将其定位。在这种处理中，使用了有限元的空间近似。我们考虑用于时间离散化的中心差分方案和用于空间变量离散化的线性有限元。示出了完全离散的方案以均方范数收敛于实际的$ H ^ 2 $解，其比率为$ C_t \ Delta t + C_s h ^ 2 / \ epsilon ^ 2 $。$ h $是网格尺寸，$ \ epsilon $是非本地交互的长度标度，$ \ Delta t $是时间步长。常量$ C_t $和$ C_s $独立于$ \ Delta t $和$ h $。在没有非线性的情况下，针对中心差时间离散化方案的能量稳定性，开发了类似于CFL的条件。作为示例，我们考虑有机玻璃并在先验误差范围内计算常数。