Researchers from the IMDEA Materials Institute and the Technical University of Madrid have developed a mathematical model that corrects the overestimated crack area obtained in computational simulations of phase-field fracture problems.

The new mathematical approach, called the Double Gradient Correction Method (DGCM), effectively addresses the overestimation of crack area in numerical simulations of material fracture.

Published in the scientific journal Computational Mechanics, this advance improves predictions of structural failure without increasing computational cost.

Phase-field-based computational models are essential tools in modern engineering for predicting how cracks propagate and how materials fail when subjected to external loads

However, conventional numerical implementations suffer from a well-known issue called strain localisation.

This numerical artifact causes the damage variable to artificially saturate across entire regions of a simulation, distorting the calculations and leading the software to predict a cracked area that is much larger than the actual one.

To correct this deviation, the DGCM method exploits a fundamental physical property, the equipartition principle of fracture energy. The study’s authors, Dr. Miguel Castillón and Professors Javier Segurado and Ignacio Romero, demonstrated that while numerical errors severely affect the direct damage variable, they leave the energy associated with its spatial gradient almost unchanged.

By duplicating the contribution associated with the spatial gradient of the damage variable, a much more accurate estimate of the fracture surface can be obtained. Validation tests showed that this approach significantly reduces the deviations found in conventional simulation methods.

“This new mathematical approach allows us to more reliably analyse the behaviour of materials near their failure limit, reducing numerical errors without adding computational complexity, and can be applied to three-dimensional models,” said Dr. Castillón.

The code required to implement this method has been released under an open-source license, facilitating its integration into standard industrial simulation tools.

The repository is hosted on GitHub and permanently archived on Zenodo. The numerical simulations were performed using ‘PhaseFieldX‘, an open-source library built upon the FEniCSx finite element framework. All results are fully reproducible with the provided materials.

Open Access funding was provided through the CRUE-CSIC agreement with Springer Nature. This work was partially supported by the Spanish Ministry of Science and Innovation through the FPI grant PRE2020-092051 (MCIN/AEI/10.13039/501100011033). J.S. and I.R. were partially supported by the Spanish Ministry of Science and Innovation through the IRIDISCENTE project (Ref. PLEC2023-010190).