  Small-scale plasticity problems are often characterised by different
patterning behaviours ranging from macroscopic down to the atomistic scale. In
successful models of such complex behaviour, its origin lies within
non-convexity of the governing free energy functional. A common approach to
solve such non-convex problems numerically is by regularisation through a
viscous formulation. This, however, may require the system to be overdamped and
hence potentially has a strong impact on the obtained results. To avoid this
side-effect, this paper addresses the treatment of the full non-convexity by an
appropriate numerical solution algorithm -- the truncated Newton method. The
presented method is a double iterative approach which successively generates
quadratic approximations of the energy landscape and minimises these by an
inner iterative scheme, based on the conjugate gradient method. The inner
iterations are terminated when either a sufficient energy decrease is achieved
or, to incorporate the treatment of non-convexity, a direction of negative
curvature is encountered. If the latter never happens, the method reduces to
Newton--Raphson iterations, solved by the conjugate gradient method, with a
subsequent line search. However, in the case of a non-convex energy it avoids
convergence to a saddle point and adds robustness. The stability of the
truncated Newton method is demonstrated for the, highly non-convex,
Peierls-Nabarro model, solved within a Finite Element framework. A potential
drop in efficiency due to an occasional near singular Hessian is remedied by a
trust region like extension, which is physically based on the Peierls-Nabarro
disregistry profile. The result is an efficient numerical scheme with a high
stability that is independent of any regularisation.
