In the paper it is proposed a new method for the solution of equilibrium problems based on a F.E. displacement formulation, that, at least in principle, is globally convergent as opposed to the classical Lagrangian method that presents only local convergence. The method, which appears to be particularly useful for plasticity models characterized by yield surfaces with regions of sharp curvature or corner points, is based on the Multiplier method. The structure of the procedure is presented and the consequent constrained optimization scheme is implemented for the case of associated plasticity coupled with damage. The main aspect of originality of the proposal is that it is not applied to the ‘return algorithm’, but to entire equilibrium iteration. At first, the local convergence properties of the constitutive equations are examined at the Gauss point level. It is proved that, also for involved constitutive models (a generalized Ottosen yield surface including isotropic hardening and damage is used in the applications), the convergence of a classical Newton’s scheme is always reached with few iterations, ensuring a quadratic rate of convergence in the solution, provided a conversion of the inequality plastic constraint into an equality one is introduced, using an augmented Lagrangian functional for exactly evaluating the slack constraints. However, it is observed that the converged stresses are often attracted far from the initial trial point, towards regions with sharper curvature, and the main reason for the lack of convergence of the procedure is found in a divergence of the solution of the non-linear equilibrium equations. It is shown that the Multiplier method allows to enlarge the radius of convergence with respect to classical iterations based on the Lagrangian method. The price for the enlargement of the convergence radius is a higher number of iterations, since the Multiplier method presents only a linear rate of convergence. Indeed, the exact fulfilment of the compatibility and admissibility equations is not attained simultaneously, once an equilibrated solution is reached, but it is iterative. In the closure of the paper a convergence analysis of an elastic–plastic problem characterized by a yield criterion resulting from the convex hull of crises surfaces, and as a consequence, presenting regions of non-differentiability, is presented. It is shown how the ability of the Multiplier method in finding the solution of the structural problem for large loading step with respect to the classical Lagrangian technique compensate its slower convergence rate.

### A globally convergent numerical algorithm for damaging elasto-plasticity based on the Multiplier method

#####
*CONTRAFATTO, Loredana Caterina;CUOMO, Massimo*

##### 2005-01-01

#### Abstract

In the paper it is proposed a new method for the solution of equilibrium problems based on a F.E. displacement formulation, that, at least in principle, is globally convergent as opposed to the classical Lagrangian method that presents only local convergence. The method, which appears to be particularly useful for plasticity models characterized by yield surfaces with regions of sharp curvature or corner points, is based on the Multiplier method. The structure of the procedure is presented and the consequent constrained optimization scheme is implemented for the case of associated plasticity coupled with damage. The main aspect of originality of the proposal is that it is not applied to the ‘return algorithm’, but to entire equilibrium iteration. At first, the local convergence properties of the constitutive equations are examined at the Gauss point level. It is proved that, also for involved constitutive models (a generalized Ottosen yield surface including isotropic hardening and damage is used in the applications), the convergence of a classical Newton’s scheme is always reached with few iterations, ensuring a quadratic rate of convergence in the solution, provided a conversion of the inequality plastic constraint into an equality one is introduced, using an augmented Lagrangian functional for exactly evaluating the slack constraints. However, it is observed that the converged stresses are often attracted far from the initial trial point, towards regions with sharper curvature, and the main reason for the lack of convergence of the procedure is found in a divergence of the solution of the non-linear equilibrium equations. It is shown that the Multiplier method allows to enlarge the radius of convergence with respect to classical iterations based on the Lagrangian method. The price for the enlargement of the convergence radius is a higher number of iterations, since the Multiplier method presents only a linear rate of convergence. Indeed, the exact fulfilment of the compatibility and admissibility equations is not attained simultaneously, once an equilibrated solution is reached, but it is iterative. In the closure of the paper a convergence analysis of an elastic–plastic problem characterized by a yield criterion resulting from the convex hull of crises surfaces, and as a consequence, presenting regions of non-differentiability, is presented. It is shown how the ability of the Multiplier method in finding the solution of the structural problem for large loading step with respect to the classical Lagrangian technique compensate its slower convergence rate.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.