The peridynamic theory reformulates the governing equation of continuum mechanics in an integro-differential form,which brings advantages in dealing with discontinuities,dynamics,and non-locality.The integro-differential formulation poses challenges to numerical solutions of complicated problems.While various numerical methods based on discretizing the computational domain have been developed and have their own merits,some important issues are yet to be solved,such as the computation of infinite domains,the treatment of softening of boundaries due to an incomplete horizon,and time error accumulation in dynamic processes.In this work,we develop the peridynamic boundary element method (PD-BEM).To this end,the boundary integral equations for static and dynamic problems are derived,and the corresponding numerical frameworks are presented.For static loading,this method gives the explicit equation solved directly without iterations.For dynamic loading,we solve the problem in the Laplace domain and obtain the results in the time domain via inversion.This treatment eliminates time error accumulation,and facilitates parallel computation.The computational results on static and dynamic examples within the bond-based peridynamic formulation exhibit several features.First,for non-destructive cases,the PD-BEM can be one to two orders of magnitude faster than the peridynamic meshless particle method (PD-MPM);second,it conserves the total energy much better than the PD-MPM;third,it does not exhibit spurious boundary softening phenomena.For destructive cases where new boundaries emerge during the loading process,we propose a coupling scheme where the PD-MPM is applied to the cracked region and the PD-BEM is applied to the un-cracked region such that the time of computation can be significantly reduced.The present method can be generalized to other subjects such as diffusion and multi-physical problems.

中文翻译：

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近场动力学的边界元法

近场动力学理论将连续介质力学的控制方程重新表述为积分微分形式，在处理不连续性、动力学和非局域性方面具有优势。积分微分公式对复杂问题的数值求解提出了挑战。而各种数值方法基于离散化的计算域已经发展起来并有自己的优点，但一些重要的问题还有待解决，例如无限域的计算，由于不完整的视界而软化边界的处理，以及动态中的时间误差累积。在这项工作中，我们开发了近场动力学边界元法（PD-BEM）。为此，推导出了静态和动态问题的边界积分方程，并给出了相应的数值框架。对于静态加载，该方法给出了直接求解的显式方程，无需迭代。对于动态加载，我们在拉普拉斯域中求解问题，并通过反演获得时域中的结果。这种处理消除了时间误差累积，便于并行计算。基于键的近场动力学公式中静态和动态示例的计算结果显示出几个特点。首先，对于非破坏性情况，PD-BEM 可以比近场动力学无网格粒子方法 (PD-MPM) 快一到两个数量级);第二，它比PD-MPM更好地保存了总能量;第三，它没有出现虚假的边界软化现象。对于在加载过程中出现新边界的破坏性情况，我们提出了一种耦合方案，其中 PD-MPM 应用于开裂区域，PD-BEM 应用于未开裂区域，这样可以显着减少计算时间。 本方法可以推广到其他主题，例如作为扩散和多物理问题。