# Convergence Check Phase-Field Scheme for Modelling of Brittle and Ductile Fractures
## 4 Convergence Check Phase-Field Algorithm
This field is read by the CCFE to check if the displacement field has obtained convergence according to the relation:  
\[\frac{\|\psi_{i}\|-\|\psi_{i-1}\|}{\|\psi_{i-1}\|}<\text{tol}. \tag{25}\]Through iterations, the displacement model also reads from the shared field of the last equilibrated \"frozen\" values of the phase-field variable \(\phi\), until criterion (25) is satisfied. During this period, to ensure that there is no change in the values of \(\phi\), stiffness matrices of the phase-field elements are diagonal matrices with an appointed diagonal value of \(10^{8}\). Once Equation (25) is satisfied, the history variable is transferred into the integration points of the second layer. The stiffness matrix of the phase-field elements takes the form of (24) and new values of the phase-field parameter are obtained and stored in the shared field. If new values of \(\phi\) in the subsequent iteration cause changes in the strain energy density which violate Condition (25), \(\phi\) is again \"frozen\" until convergence of the displacement field. In every iteration, the CCFE also checks if convergence of the phase-field model has been obtained by the relation:  
\[\frac{\left\|\phi_{i}\right\|-\left\|\phi_{i-1}\right\|}{\left\|\phi_{i-1} \right\|}<\text{tol.} \tag{26}\]  
However, Criterion (26) is not sufficient by itself for further actions in seeking the convergence. Only when both (25) and (26) satisfy their tolerance is a further check of the convergence conducted. If Condition (26) is satisfied (and (25)), the \(L^{2}\) norm of the converged phase-field distribution is stored as \(\left\|\phi_{k}\right\|^{c}=\left\|\phi_{i}\right\|\), required for control of the convergence of the coupled problem (displacement and phase field). Convergence of the coupled problem is user controlled by controlling the nodal force vector of the CCFE, where the stiffness matrix of the CCFE is again a diagonal matrix with an appointed value \(10^{8}\). Coupled convergence is checked after a minimum of five iterations, which are required for the exchange of variables between layers and confirmation of convergence of each layer separately. Therefore, early in the fifth iteration, if Criterions (25) and (26) are satisfied, the CCFE controls the criterion:  
\[\frac{\left\|\phi_{k}\right\|^{c}-\left\|\phi_{k-1}\right\|^{c}}{\left\|\phi_{ k-1}\right\|^{c}}<\text{tol.} \tag{27}\]  
Until satisfaction of Condition (27), the nodal force vector of the CCFE is the unit vector, preventing ABAQUS from pronuncing convergence and forcing additional iterations, even  
Figure 2: Scheme of the CCPF algorithm. if default ABAQUS convergence criterions are satisfied. In all examples in this paper, the tolerance parameter is set to \(\text{tol}=5\cdot 10^{-5}\) for all three additional convergence conditions. As can be seen, to pronounce convergence of a coupled problem for a corresponding loading increment, local convergence of both layers should be reached, firstly according to the ABAQUS convergence conditions and secondly according to Equations (25) and (26). When additional convergence criterions are satisfied, Equation (27) ensures that relative change between converged states of the phase-field layer is less than the required tolerance. In a physical sense, Condition (27) ensures that for a corresponding loading increment, a global minimum of a two-layer coupling is found and a sufficient number of iterations is calculated, where any potential crack initiation of propagation is captured. ABAQUS subroutines of the CCPF algorithm for the problem of brittle fractures are available for download to all interested readers, where ABAQUS.inp and.for files are provided for the problem of a single-edge notched plate loaded onto tension.