  We develop an approximation scheme for three-dimensional dislocation dynamics
in which the dislocation line density is concentrated at points, or monopoles.
Every monopole carries a Burgers vector and an element of line. The monopoles
move according to mobility kinetics driven by elastic and applied forces. The
divergence constraint, expressing the requirement that the monopoles
approximate a boundary, is enforced weakly. The fundamental difference with
traditional approximation schemes based on segments is that in the present
approach an explicit linear connectivity, or 'sequence', between the monopoles
need not be defined. Instead, the monopoles move as an unstructured point set
subject to the weak divergence constraint. In this sense, the new paradigm is
'line-free', i. e., it sidesteps the need to track dislocation lines. This
attribute offers significant computational advantages in terms of simplicity,
robustness and efficiency, as demonstrated by means of selected numerical
examples.
