Acknowledgements

• This work was supported by the Spanish Ministry of Science and Innovation through the FPI grant PRE2020-092051 (MCIN/AEI/10.13039/501100011033).

• 13. Castillón gratefully acknowledges Dr. Jørgen S. Dokken for his invaluable technical assistance with the implementation of Lagrange multiplier methods in the FEniCSx computational framework. His guidance proved essential for the development of the energy-controlled solvers presented in this work and their subsequent integration into the PhaseFieldX library (Castillón[1]).

[1]

M. Castillón. Phasefieldx: an open-source framework for advanced phase-field simulations. Journal of Open Source Software, 10(108):7307, 2025. doi:https://doi.org/10.21105/joss.07307.