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Motivated by the inadequacy of conducting atomistic simulations of crackpropagation using static boundary conditions that do not reflect the movementof the crack tip, we extend Sinclair's flexible boundary condition algorithm[Philos. Mag. 31, 647-671 (1975)] and propose a numerical-continuation-enhancedflexible boundary (NCFlex) scheme, enabling full solution paths for cracks tobe computed with pseudo-arclength continuation, and present a method forincorporating more detailed far-field information into the model for next to noadditional computational cost. The new algorithms are ideally suited to studydetails of lattice trapping barriers to brittle fracture and can beincorporated into density functional theory and multiscale quantum/classicalQM/MM calculations. We demonstrate our approach for Mode III fracture with a 2Dtoy model and mploy it to conduct a 3D study of Mode I fracture of siliconusing realistic interatomic potentials, highlighting the superiority of the newapproach over employing a corresponding static boundary condition. Inparticular, the inclusion of numerical continuation enables converged resultsto be obtained with realistic model systems containing a few thousand atoms,with very few iterations required to compute each new solution. We alsointroduce a method to estimate the lattice trapping range of admissible stressintensity factors $K_- < K < K_+$ very cheaply and demonstrate its utility onboth the toy and realistic model systems.

A numerical-continuation-enhanced flexible boundary condition scheme applied to Mode I and Mode III fracture