## 5 Proposed Predictor-Corrector Implicit to Explicit  
Many authors have used UMAT user-subroutines to implement new materials in ABAQUS without resourcing a fully Jacobi stiffness matrix (aka tangent matrix), using only a secant matrix for the ABAQUS predictor-corrector, with viscous regularization. As referenced before, viscous regularization has only been demonstrated to be a fully implicit method when a tangent matrix is applied to the predictor-corrector algorithm [54; 56]. The use of a secant matrix with viscous regularization lacks the theoretical background to prove its implicit convergence. It can be depicted in expression (42) that the if the time step is larger than the viscosity parameter \(\Delta t\gg\eta\), then the level of regularization is low enough, and the implicit standard convergence can be admitted since the viscous damage is always related to the current time step as the non-viscous damage. However, if the time step is lower than the viscosity parameter \(\Delta t\ll\eta\), then the viscous damage will always be related to the previous time step, therefore prompting an explicit time step integration. Conflicts may arise in this last scenario because the strain increment corresponds to a full implicit integration strategy (45). These inconsistencies happen when damage follows an explicit time step update \(d_{v}^{t+\Delta t}\simeq d_{v}^{t}\), and the strain increment follows a full implicit integration scheme. This can cause some convergence problems, especially if the time step is very small \(\Delta t\ll\eta\). Furthermore, when the time step is very small since the viscous damage solution overrides the non-viscous damage solution, the viscous damage energy tends to grow (44), indirectly providing extra energy that is not real. \[\epsilon_{t+\Delta t}=\epsilon_{t}+\Delta\epsilon_{t+\Delta t} \tag{45}\]  
To solve this problem, a new implicit-to-explicit algorithm is proposed. The fundamental basis guarantees that when the time step is large, the damage evolution follows an implicit time step integration, but when the time step is smaller than the viscosity parameter, the damage evolution switches to an explicit time step integration. This is an optimization of the predictor-corrector of ABAQUS, which states that when the time step is large a full implicit formulation is used, but when the predictor-corrector algorithm of ABAQUS presents complications in converging or reducing the time step, then an explicit formulation is used to suppress the converging difficulty. The basis of this formulation is using the increment correction parameter \(\varpi\) linked to the portion of the non-viscous damage variable \(d^{t+\Delta t}\) in expression (46). \[\epsilon_{t+\Delta t}=\epsilon_{t}+\varpi\times\Delta\epsilon_{t+\Delta t} \tag{46}\]  
in which:  
\[\varpi=\frac{\Delta t}{\Delta t+\eta} \tag{47}\]It is important to point out that when \(\varpi=1.0\) the material stiffness and stress update during the time integration in the UMAT follows a full implicit formulation; in contrast, when \(\varpi=0.0\), it follows a full explicit time integration for the material stiffness and stress update. Using this algorithm, it is guaranteed that when performing the viscous regularization, if the predictor-corrector uses large time steps, then both the viscous and non-viscous solution portion of expression (42) will contribute to the implicit time step integration. However, when the time step is small enough, then the increment correction parameter tends to be small, close to an explicit integration, and the influence of the non-viscous solution overrides the viscous solution, therefore using only the previous damage variable. This way there is no mixed implicit time step integration, with previous step damage variables, as suggested before, perhaps compromising the convergence and accuracy of the numerical solution.