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Motivated by the inadequacy of conducting atomistic simulations of crack propagation using static boundary conditions that do not reflect the movement of the crack tip, we extend Sinclair's flexible boundary condition algorithm [Philos. Mag. 31, 647-671 (1975)] and propose a numerical-continuation-enhanced flexible boundary (NCFlex) scheme, enabling full solution paths for cracks to be computed with pseudo-arclength continuation, and present a method for incorporating more detailed far-field information into the model for next to no additional computational cost. The new algorithms are ideally suited to study details of lattice trapping barriers to brittle fracture and can be incorporated into density functional theory and multiscale quantum/classical QM/MM calculations. We demonstrate our approach for Mode III fracture with a 2D toy model and mploy it to conduct a 3D study of Mode I fracture of silicon using realistic interatomic potentials, highlighting the superiority of the new approach over employing a corresponding static boundary condition. In particular, the inclusion of numerical continuation enables converged results to be obtained with realistic model systems containing a few thousand atoms, with very few iterations required to compute each new solution. We also introduce a method to estimate the lattice trapping range of admissible stress intensity factors $K_- < K < K_+$ very cheaply and demonstrate its utility on both the toy and realistic model systems.