# The Boundary Element Method of Peridynamics

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.

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