Given the new value of ${\dot{\epsilon }}^p$ and ${\epsilon }^p$, and the old value $X_d$, $X_s$, $X_g$, ${\rho }_i$, g and $c_v$.

Update ${\rho }_i$ and $c_v$ with the non-linear solution of Equations (9) and (21). The non-linear solution for the set of equations is carried out a Newton-Raphson scheme. Calculate the updated value of the grain size g with Equation (13). Calculate the grain grow rate with the old, the new value of the grain size and the size of the time step. Update $X_s$ with Equation (19), $X_d$ with Equation (20) and $X_g$ with Equation (18). Calculate the long-range component of the flow stress using Equation (5).