The peridynamic theory reformulates the equations of continuum mechanics in terms of integro-differential equations instead of partial differential equations. It is not trivial to directly apply naive approach in artificial boundary conditions for continua to peridynamics modeling, because it usually involves semi-discretization scheme. In this paper, we present a new way to construct exact boundary conditions for semi-discretized peridynamics using kernel functions and recursive relations. Specially, kernel functions are used to characterize one single source are combined to construct the exact boundary conditions. The recursive relationships between the kernel functions are proposed, therefore the kernel functions can be computed through the ordinary differential system and integral system with high precision. The numerical results demonstrate that the boundary condition has high accuracy. The proposed method can be applied to modeling of wave propagation of other nonlocal theories and high dimensional cases.

中文翻译：

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一维半离散近场动力学的精确人工边界条件

近场动力学理论根据积分微分方程而不是偏微分方程重新表述连续介质力学方程。直接在人工边界条件中应用朴素的方法来进行近场动力学建模的连续性并非易事，因为它通常涉及半离散化方案。在本文中，我们提出了一种使用核函数和递归关系为半离散近场动力学构建精确边界条件的新方法。特别地，核函数被用来表征一个单一的来源，并结合起来构造精确的边界条件。提出了核函数之间的递推关系，从而可以通过普通微分系统和积分系统高精度地计算核函数。数值结果表明边界条件具有较高的精度。所提出的方法可以应用于其他非局部理论和高维情况的波传播建模。