Augmented Lagrangian Treatment of Inequality Constraints in Phase-Field Modelling of Microstructure Evolution

In this work, a phase-field framework is implemented to investigate the microstructure evolution in Ti-Mo systems involving the parent $\beta$ phase and four $\omega$ variants [1], with particular emphasis on the high $\omega$–$\omega$ interfacial energy, which leads to separation of $\omega$ variants by the $\beta$ phase. Each phase is characterized by an order parameter constrained between 0 and 1, with all the order parameters satisfying the sum-to-unity constraint. The microstructure evolution is governed by the minimization of a global rate-potential [2], with interfacial energy contributions characterized through a double-obstacle potential [3]. To evaluate computational performance and accuracy, the numerical behavior of this formulation is benchmarked against a multi-well double-ditch potential formulation within the governing phase-field framework [4]. While the constraints can be enforced by a standard penalty formulation, modeling the high-energy $\omega$–$\omega$ interfaces remains challenging because it requires very large penalty parameters, which lead to ill-conditioning and inefficient performance of iterative solvers. To overcome this limitation, an augmented Lagrangian framework is implemented in the form of the Uzawa algorithm, where the primal variables are solved for fixed dual variables and the dual variables are then updated systematically. The effectiveness of this approach is demonstrated through preliminary one-dimensional simulations. Subsequently, the proposed formulation is scaled to higher-dimensional simulations, emphasizing its robustness and improved convergence behavior in complex scenarios.

References

1. S. Banerjee and P. Mukhopadhyay, Phase transformations: examples from titanium and zirconium alloys, Elsevier, 12, 2010.
2. C. Miehe, A multi-field incremental variational framework for gradient-extended standard dissipative solids, Journal of the Mechanics and Physics of Solids, 59(4):898–923, 2011.
3. I. Steinbach, Phase-field models in materials science, Modelling and simulation in materials science and engineering, 17(7):073001, 2009.
4. J. Kozlík, K. Tůma, O. Souček, J. Dobrzański, and S. Stupkiewicz, Multiwell phase-field model for arbitrarily strong total-spreading case, International Journal of Engineering Science, 221:104474, 2026.