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A method for solving general boundary-value problems involving discrete dislocations is introduced. Plastic flow emerges from the motion of dislocations in an incremental fashion. At each increment, the displacement, strain and stress fields in the body are obtained by superposition of the infinite medium fields associated with individual dislocations and an image field that enforces boundary conditions. Dislocations are represented as monopoles and dislocation events are treated as a transportation map problem. Long-range interactions are accounted for through linear elasticity with a core regularization procedure. At the current state of development of the method, no ad hoc short-range interactions are included. An approximate loop nucleation model is used for large-scale computations. The image problem is solved using a finite element formulation with the following features: (i) a single Cholesky decomposition of the global stiffness matrix, (ii) a consistent enforcement of traction and displacement boundary conditions, and (iii) image force interpolation using an efficient BB-tree algorithm. To ensure accuracy, we explore stable time steps and employ monopole splitting techniques. Special attention is given to the interaction of curved dislocations with arbitrary domain boundaries and free surfaces. The capabilities of the framework are illustrated through a wire torsion problem.