  We present a continuum model to determine the dislocation structure and
energy of low angle grain boundaries in three dimensions. The equilibrium
dislocation structure is obtained by minimizing the grain boundary energy that
is associated with the constituent dislocations subject to the constraint of
Frank's formula. The orientation-dependent continuous distributions of
dislocation lines on grain boundaries are described conveniently using the
dislocation density potential functions, whose contour lines on the grain
boundaries represent the dislocations. The energy of a grain boundary is the
total energy of the constituent dislocations derived from discrete dislocation
dynamics model, incorporating both the dislocation line energy and reactions of
dislocations. The constrained energy minimization problem is solved by the
augmented Lagrangian method and projection method. Comparisons with atomistic
simulation results show that our continuum model is able to give excellent
predictions of the energy and dislocation densities of both planar and curved
low angle grain boundaries.
