# Convergence Check Phase-Field Scheme for Modelling of Brittle and Ductile Fractures
## 4 Convergence Check Phase-Field Algorithm  
The convergence check phase-field (CCPF) algorithm is unlike the many solution algorithms proposed in recent years for dealing with phase-field methodology. Within this manuscript, the proposed methodology references the RCTRL algorithm [27], emphasising the benefits of the newly proposed scheme, since the co-authors of the paper were involved in the derivation and application of the aforementioned framework. One great distinction is the number of mesh layers. In the CCPF algorithm, two meshes are sufficient (displacement and phase field), as in most staggered frameworks. As the name states, the CCPF algorithm relies on the insurance of the convergence of the first mesh layer prior to the transfer of the corresponding variable to the second layer. Basically, during finding the equilibrium in one mesh layer, the solution variable of the other mesh is kept \"frozen\", i.e., there is no update to the solution variables in the other layer. This \"freezing\" approach is standard in the return-mapping algorithm for plasticity problems. A similar approach was proposed in the phase-field methodology in [29] for solving brittle and ductile damage. In this paper, the CCPF framework based on the \"freezing\" of corresponding solution variables is implemented in commercial FE code ABAQUS through user subroutines for solving problems of brittle and ductile softening in monotonic and cyclic loading regimes (fatigue). For the problem of high-cyclic fatigue, a cycle-skipping scheme was utilised. In the first mesh layer, a UMAT subroutine in combination with the ABAQUS built-in finite elements was employed for the determination of stress, strain, and displacement distribution, allowing implementation of the strain energy density decomposition, required for obtaining physically consistent crack initiation and propagation. The UEL subroutine is employed for implementation of the phase-field in the second mesh layer. More precisely, the first layer consists of quadrilateral CPE4 finite elements, adopting the plane strain assumption. Accordingly, the second mesh layer consists of quadrilateral phase-field user-defined finite elements. The lack of staggered phase-field algorithms creates a need for many loading increments to ensure the correct numerical solution. In the RCTRL scheme, this issue is overwhelmed through application of the third mesh layer consisting of the phase-field elements, allowing the application of large loading increments. Instead of a whole layer of elements, in the CCPF framework, a single finite element, namely, the \"convergence check finite element\" (CCFE) is employed, thus reducing numerical complexity for one mesh layer compared to the RCTRL scheme. Based on the finite element formulation of the two main mesh layers, CCPF is derived as a quadrilateral finite element. The CCFE checks for the convergence of both mesh layers for variable exchange purposes and controls whether an energy minimum is reached for the appointed loading increment (global equilibrium). The convergence check element is also embedded by the UEL subroutine. The scheme of the CCPF algorithm is presented in Figure 2.  
As presented in Figure 2, any problem solved by the CCPF algorithm consists of two mesh layers and one CCFE finite element. It is important to mention that in the CCPF framework, the ABAQUS solver and its convergence measurement tools are utilised, as in usual numerical analysis. On top of the standard ABAQUS convergence checking criterions, the following text explains additional convergence check tools of the CCPF scheme, which ensure the application of larger loading increments. After imposing a loading increment, the first step is finding the equilibrium of the displacement field in the first layer. Through all integration points, for iteration \(i\), strain energy density \(\psi\) is stored in a shared field.