A primal–dual interior point method to implicitly update Gurson–Tvergaard–Needleman model

Abstract

This study proposes an implicit algorithm applying the primal–dual interior point method (PDIP method) to stabilize the stress update when using a class of the Gurson–Tvergaard–Needleman model (GTN model). The GTN model is widely used to realize the change in void volume fraction that governs ductile fracture in metals, but numerical instabilities arise due to shrinkage of the yield surface and the accelerated void growth. In fact, such shrinkage can lead to misjudgment of yield conditions when using conventional return mapping algorithms, since trial elastic stresses are computed assuming zero incremental plastic strain. In addition, the change in void volume fraction is often approximated in bilinear form to represent the acceleration of void growth, but should be smooth to apply nonlinear solution methods such as the Newton’s method. To avoid such inconvenience in the implicit stress update for the GTN model and ensure numerical stability, we propose an algorithm that replaces the constitutive equations with inequality constraints with an equivalent constrained optimization problem by applying the PDIP method. After verifying the numerical accuracy and convergence of the proposed implicit algorithm using iso-error maps, we demonstrate its capability through several numerical examples that cannot be solved by the conventional return mapping algorithm or the PDIP method applied only to the inequality constraint corresponding to the yield condition.

Formulation for irreversible damage

As in Sect. 5.1, the maximization problem (63) subject to the inequality constraints (64) and (65) can be rewritten as

$$\begin{aligned} \underset{ \left\{ \varvec{\tau }, \; Y_i\; A \right\} }{\overset{}{\text {maximize}} }&\quad \varvec{\tau } :\tilde{\varvec{d}}^\textrm{p} + \sum _{i=1}^{2} Y_{i} \dot{f}_{i} - A \dot{\alpha } \nonumber \\&\quad + \mu ^\textrm{p}\log {s^\textrm{p}} + \sum _{i=1}^{2} \mu ^{\textrm{f}}_i\log {s^{\textrm{f}}_i} \end{aligned}$$

(A1)

$$\begin{aligned} \mathrm {subject\,to}&\quad \Phi ^\textrm{p}(\varvec{\tau },A) + s^\textrm{p} = 0 \end{aligned}$$

(A2)

$$\begin{aligned}&\quad \Phi ^{\textrm{f},(i)}_\mathrm {\sigma }(Y_i) + s^{\textrm{f}}_i = 0 \ (i=1, \, 2). \end{aligned}$$

(A3)

where $\mu _{1}^\textrm{p}$ and $\mu ^{\textrm{f}}_i$ are the duality gaps, and $s^\textrm{p} > 0$ and $s^{\textrm{f}}_i > 0$ are the slack variables. Then, the Lagrangian functional is defined as

$$\begin{aligned}&\mathcal {L}_\textrm{GTN}^\textrm{IP} \left( \varvec{\tau }, Y_i, A, \dot{\lambda }^\textrm{p}, \dot{\lambda }^\textrm{f}_i \right) \nonumber \\&\qquad = \varvec{\tau } : \tilde{\varvec{d}}^\textrm{p} + \sum _{i=1}^{2} Y_{i} \dot{f}_{i} - A \dot{\alpha } \nonumber \\&\qquad \ \ + \mu ^\textrm{p}\log {s^\textrm{p}} + \sum _{i=1}^{2} \mu ^{\textrm{f}}_i\log {s^{\textrm{f}}_i} \nonumber \\&\qquad \ \ -\dot{\lambda }^\textrm{p} \left( \Phi ^\textrm{p} (\varvec{\tau }, A)+s^\textrm{p}\right) \nonumber \\&\qquad \ \ -\sum _{i=1}^{2} \dot{\lambda }^\textrm{f}_i \left( \Phi ^{\textrm{f},(i)}_\mathrm {\sigma } (Y_i)+ s^{\textrm{f}}_i\right) . \end{aligned}$$

(A4)

where $\dot{\lambda }^\textrm{p}$ and $\dot{\lambda }^\textrm{f}_i$ are the Lagrange multipliers. The stationary point of this Lagrangian function can be achieved as

In the numerical algorithm that was presented in Sect. 5.3, the duality gaps $\mu ^\textrm{p}$ and $\mu ^{\textrm{f}}_i$ are gradually brought close to zero by the path-following method to obtained the above saddle point, while Eqs. (A5)–(A11) are numerically solved by the Newton’s method.