The numerical solution of peridynamics equations is usually done by using uniform spatial discretisation. Although implementation of uniform discretisation is straightforward, it can increase computational time significantly for certain problems. Instead, non-uniform discretisation can be utilised and different discretisation sizes can be used at different parts of the solution domain. Moreover, the peridynamic length scale parameter, horizon, can also vary throughout the solution domain. Such a scenario requires extra attention since conservation laws must be satisfied. To deal with these issues, dual-horizon peridynamics was introduced so that both non-uniform discretisation and variable horizon sizes can be utilised. In this study, dual-horizon peridynamics formulation is derived by using Euler–Lagrange equation for state-based peridynamics. Moreover, application of boundary conditions and determination of surface correction factors are also explained. Finally, the current formulation is verified by considering two benchmark problems including plate under tension and vibration of a plate.

通常通过使用统一的空间离散化来完成周动力学方程的数值解。尽管统一离散化的实现很简单，但是对于某些问题，它可以大大增加计算时间。相反，可以使用非均匀离散化，并且可以在解决方案域的不同部分使用不同的离散化大小。此外，在整个求解域中，围动力学长度尺度参数（horizo​​ntal）也可以变化。由于必须遵守保护法律，因此需要格外注意。为了解决这些问题，引入了双水平周动力学，以便可以利用非均匀离散和可变视界大小。在这项研究中，使用基于状态的动力学的欧拉-拉格朗日方程推导了双水平周动力学公式。此外，还说明了边界条件的应用和表面校正因子的确定。最后，通过考虑两个基准问题来验证当前的公式，这些问题包括板在张力下的振动和板的振动。