We seek the solution for which the\(\mathbf{r}_{I}^{\mathbf{u}}\mathbf{=0}\) and\(\mathbf{r}_{I}^{\varphi}\mathbf{=0}\). Due to the nonlinear nature of the residuals with respect to \(\mathbf{u}\) and \(\varphi\), we employ an incremental-iterative strategy utilizing the Newton–Raphson approach in conjunction with a parametrization based on a fictitious time: