Levenberg–Marquardt vs Powell’s dogleg method for Gurson–Tvergaard–Needleman plasticity model

► GTN equations are viewed as a non-linear optimization problem. ► The Levenberg–Marquardt method is more robust than Powell’s dogleg for GTN equations. ► The Levenberg–Marquardt method is preferable for implicit FE solvers of GTN model. The GTN continuous damage model is very popular in academia and industry for structural integrity assessment and ductile fracture simulation. Following Aravas’ influential 1987 paper, Newton’s method has been used widely to solve the GTN equations. However, if the starting point is far from the solution, then Newton’s method can fail to converge. Hybrid methods are preferred in such cases. In this work we translate the GTN equations into a non-linear minimization problem and then apply the Levenberg–Marquardt and Powell’s ‘dogleg’ hybrid methods to solve it. The methods are tested for accuracy and robustness on two simple single finite element models and two 3D models with complex deformation paths. In total nearly 137,000 different GTN problems were solved. We show that the Levenberg–Marquardt method is more robust than Powell’s method. Our results are verified against the Abaqus’ own solver. The superior accuracy of the Levenberg–Marquardt method allows for larger time increments in implicit time integration schemes.