Application of a modified semismooth Newton method to some elasto‐plastic problems

Some elasto‐plasticity models with hardening are discussed and some incremental finite element methods with different time discretisation schemes are considered. The corresponding one‐time‐step problems lead to variational equations with various non‐linear operators. Common properties of the non‐linear operators are derived and consequently a general problem is formulated. The problem can be solved by Newton‐like methods. First, the semismooth Newton method is analysed. The local superlinear convergence is proved in dependence on the finite element discretisation parameter. Then it is introduced a modified semismooth Newton method which contain suitable ‘damping’ in each Newton iteration in addition. The determination of the damping coefficients uses the fact that the investigated problem can be formulated as a minimisation one. The method is globally convergent, independently on the discretisation parameter. Moreover the local superlinear convergence also holds. The influence of inexact inner solvers is also discussed. The method is illustrated on a numerical example.