Application and adaptation of the truncated Newton method for non-convex plasticity problems

Abstract

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.