# Convergence Check Phase-Field Scheme for Modelling of Brittle and Ductile Fractures
## 6 Conclusions  
The paper proposes a novel convergence check phase-field staggered scheme for fracture modelling using a phase-field method. This novel methodology can be considered as a continuation of the previously published work related to the phase-field method, where the three-layer RCTRL scheme was proposed and demonstrated as very efficient, alleviating the usual drawbacks of staggered schemes. In this work, the computational burden caused by three dense mesh layers is avoided, since only a single, user-defined convergence check finite element (CCFE) checks for the global convergence of the coupled problem and replaces the third mesh of the RCTRL scheme. Aside from controlling global convergence, the CCFE also controls solution variables passing from one mesh layer to another. Unlike most staggered frameworks which pass solution variables between layers in every iteration, in the CCPF algorithm, the strain energy density history variable is passed to the phase-field layer only when convergence of the displacement layer is satisfied. The proposed methodology is derived from the adopting plane strain and 3D setting. The efficiency of the algorithm was demonstrated by comparison with the RCTRL scheme, where the CCPF algorithm has shown faster computational time and a substantially smaller number of iterations required during crack evolution. The CCPF scheme was implemented into FE software ABAQUS by means of the user subroutines UMAT and UEL. The framework was tested by solving problems of brittle and ductile fractures in monotonic loading regimes. Most of the examples assume a small-strain setting, but it has been demonstrated that the ABAQUS extension towards large-strain theory can be easily applied using the NLGEOM option, giving more accurate results for the ductile damage phenomena. Aside from monotonic loading, validity of the CCPF scheme was tested on problems dealing with  
Figure 22: Fatigue crack propagation in the HCF regime through cycles: (**a**) 26,000 cycles, (**b**) 91,000 cycles, (**c**) 141,000 cycles, (**d**) 195,000 cycles.  
low and high cyclic fatigue. An increase in speed for simulations of fatigue problems was provided by the cycle-skipping technique relying on the extrapolation of the fatigue variable, embedded by means of the user subroutine UEXTERNALDB. Additionally, the CCPF framework utilises ABAQUS built-in advanced options, such as line search for convergence improvement and element deletion for better visualization of the damage and numerical efficiency. All interested readers can download routines (.for file) of the CCPF scheme and cycle-skipping technique proposed by the authors and.inp files of the benchmark examples.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app13137776/s1](https://www.mdpi.com/article/10.3390/app13137776/s1)]([https://www.mdpi.com/article/10.3390/app13137776/s1](https://www.mdpi.com/article/10.3390/app13137776/s1)). File S1: CT-HCF.for, File S2: CT-HCF.inp, File S3: Tension.for, File S4: Tension.inp.  
T.L.: Conceptualization, Methodology, Software, Validation, Writing--original draft preparation, Investigation; T.P.: Formal analysis, Data curation, Visualization, Writing--review and editing; Z.T.: Resources, Supervision, Project administration, Funding acquisition. All authors have read and agreed to the published version of the manuscript.  
This work was supported and co-funded by the European Union through the European Regional Development Fund, Operational Programme \"Competitiveness and Cohesion 2014-2020\" of the Republic of Croatia, project ImproWE--Improvement of High-efficiency Welding Technology (KK.01.1.1.07.0075).  
Not applicable.  
Data Availability Statement: Data is contained within the article or Supplementary Materials.  
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.