# Convergence Check Phase-Field Scheme for Modelling of Brittle and Ductile Fractures
## 4 Convergence Check Phase-Field Algorithm
### Cycle-Skipping Technique  
The phase-field formulation is computationally demanding due to the requirement of dense mesh discretisation which exhibits lengthy computational time. In the case of solving high-cyclic fatigue phenomena, these requirements are multiplied by the number of simulated loading cycles. Therefore, a two-step cycle-skipping technique is embedded into existing numerical routines for the phase-field formulation, relying on the energy accumulation variable \(\overline{\psi}\). The first step of skipping occurs during the initial degradation of the bulk material, according to the relation:  
\[F\big{(}\overline{\psi}\big{)}\psi_{\text{c}}<\psi. \tag{28}\]  
Prior to damage initiation, constant loading amplitude leads to constant change in the strain energy density \(\Delta\psi\). This can be written as:  
\[\overline{\psi}=\text{N}_{1}\Delta\psi. \tag{29}\]  
For the known fatigue degradation function, (in this paper \(\hat{F}_{\text{HCF}}\big{(}\overline{\psi}\big{)}\)), by inserting (29) into Equation (28), the number of cycles that can be skipped during the first step is computed as:  
\[\text{N}_{1}=\frac{\overline{\psi}_{\infty}}{\Delta\psi}10^{-\sqrt{\frac{ \overline{\psi}}{\psi_{\text{c}}}}} \tag{30}\]  
To ensure there is no degradation of the material after cycle skipping according to Equation (30), from \(\text{N}_{1}\) cycles which can be skipped, few of them can be subtracted. In this paper, the number of subtracted cycles is two, while the number of jumped cycles \(\text{N}_{1}\) is calculated after the first loading increment. Afterwards, the smallest number of cycles \(\text{N}_{1}\) is used as multiplication factor in accordance with Equation (29) and is imposed in the first unloading increment. After the onset of damage, the second cycle-skipping step is performed. It is based on the extrapolation procedure proposed in [62] and modified for application in the phase-field method in [48]. After established N loading cycles and fatigue variable \(\overline{\psi}(\text{N})\) for two consecutive loading cycles exhibiting change in the accumulation variables \(\Delta\overline{\psi}_{1}\) and \(\Delta\overline{\psi}_{2}\), respectively, the number of skipped and extrapolated cycles can be computed by the relation:  
\[\Delta\text{N}_{2}=q\frac{\Delta\overline{\psi}_{1}}{\Delta\overline{\psi}_{1 }-\Delta\overline{\psi}_{2}} \tag{31}\]  
where \(q\) represents a fidelity parameter, which controls the maximum number of jumped cycles allowed. After finding the appropriate number of skipped cycles according to Equation (31), the result is used for extrapolation of the accumulation variable onto a previously achieved value after N cycles as:  
\[\overline{\psi}(\text{N}+\Delta\text{N}_{2})=\overline{\psi}(\text{N})+ \Delta\overline{\psi}_{1}\Delta\text{N}_{2}+\frac{1}{2}\big{(}\Delta\overline {\psi}_{1}-\Delta\overline{\psi}_{2}\big{)}(\Delta\text{N}_{2})^{2} \tag{32}\]as represented in Figure 3.  
The presented two-step procedure is embedded into FE software ABAQUS by means of the subroutine UEXTERNALDB. After first stage of cycle jumping, according to the procedure proposed by the authors, the simulation uses self-checking in a loop to determine whether there is a possibility to extrapolate some loading cycles after the user-controlled number of cycles nFE are computed one-by-one. In this paper, the extrapolation check is performed after every 10 loading cycles. Obviously, the choice of the fidelity parameter \(q\) and parameter nFE affects computational time as correctness of the numerical solutions obtained, and therefore, they should be taken with caution. The ABAQUS subroutine of the CCPF algorithm with the cycle-skipping technique is provided by the authors for download, where the problem of high-cyclic fatigue on the CT specimen can be assessed. This is also discussed later in the paper.