## 4 Finite Element Model and Solution of the Nonlinear Discrete Equations
### A Specialized Incremental-Iterative Solution  
A specialized numerical solution strategy is now presented, similar to that used for the evaluation of the equilibrium path of geometrically nonlinear structures. It will be used to evaluate the deformed configuration of the pipeline once the assigned generalized strains \(\bar{\mathbf{\rho}}\) are known. Once the standard finite element assembly procedures have been carried out, the equilibrium configuration for assigned strains is obtained from the solution of the following system of nonlinear algebraic equations  
\[\mathbf{s}(\mathbf{d})-\mathbf{p}=\mathbf{0} \tag{27}\]  
where \(\mathbf{p}\) and \(\mathbf{s}\) represent, respectively, the known vector due to the measured strains and the internal forces vector, which are obtained by means of finite element assemblages of the quantities reported in Equation (24). Vector \(\mathbf{d}\) collects all the DOFs of the problem. To make it possible the solution of the nonlinear system in Equation (27) by means of an incremental formulation based on the Newton method the problem is reformulated as  
\[\mathbf{s}(\mathbf{d})-t\mathbf{p}=\mathbf{0} \tag{28}\]  
with the introduced load factor \(t\) which assumes values in \([0,1]\). In this way it is possible to evaluate a sequence of equilibrium points \(\mathbf{z}^{(k)}\equiv\{\mathbf{d}^{(k)}\), \(t^{(k)}\}\) associated to increasing values of \(t\), that is \(t^{(k)}\leq t^{(k+1)}\), in order to gradually reach the value of \(t=1\). This allows exploiting the properties of the incremental strategy based on the [PERSON] method. The strategy so proposed is then similar to that used to evaluate the equilibrium path of slender elastic structures in case of large deformations and applied external loads [12]. For the problem at hand, since the equilibrium path does not present a limit point for the parameter \(t\), it is possible to use a \(t\)-controlled scheme. Assuming a sequence of increasing values of the control parameter \(t=t^{(k)}\) the solution of the nonlinear system  
\[\mathbf{r}(\mathbf{d},t^{(k)})\equiv\mathbf{s}(\mathbf{d})-t^{(k)}\mathbf{p}= \mathbf{0} \tag{29}\]  
defines points (steps) \(\mathbf{z}^{(k)}\equiv\{\mathbf{d}^{(k)}\), \(t^{(k)}\}\) which satisfy the problem Equation (28). Starting from a known point \(\mathbf{z}_{0}\equiv\mathbf{z}^{(k)}\) which fulfills Equation (29), the first step of the method to evaluate a new point \(\mathbf{z}^{(k+1)}\) is to obtain a _predictor_\(\mathbf{z}_{1}=\{\mathbf{d}_{1}\), \(t^{(k)}\}\) using an extrapolation of the previous evaluated point \(\mathbf{d}_{1}=\mathbf{d}^{(k)}+\beta(\mathbf{d}^{(k)}-\mathbf{d}^{(k-1)})\) with \(\beta\in\mathbb{R}\) a real number which gives the step amplitude. \(\mathbf{d}_{1}\) is evaluated along the initial path tangent for \(k=1\). The predictor value is subsequently corrected by means of a sequence of estimates \(\mathbf{z}_{j}\) using a Newton-Raphson iteration  
\[ \begin{cases}\mathbf{K}_{t}\dot{\mathbf{d}}=-\mathbf{r}_{j}\\ \mathbf{d}_{j+1}=\mathbf{d}_{j}+\dot{\mathbf{d}}\end{cases}  \tag{30}\]  
where \(\mathbf{K}_{t}\) is the Jacobian of the nonlinear system of Equation (29) evaluated in \(\mathbf{d}_{j}\)  
\[\mathbf{K}_{t}=\left.\frac{\partial\mathbf{s}(\mathbf{d})}{\partial\mathbf{d} }\right|_{\mathbf{d}_{j}}.\]  
obtained by assembling the element contributions in Equation (25). It is worth noting that, due to the high efficiency and robustness of the mixed (stress-displacement) iterations, the method usually converges in a single increment, i.e., the solution is possible directly for \(t=1\). The possibility of sub-increments is however maintained for more complex situations with respect to the applications considered in this paper. Solution in Condensed Form  
The system in Equation (30) has a block decoupling due to the local nature of variables \(\mathbf{t}_{g}\) which are defined at the element Gauss point level and do not require the imposition of continuity as occurs for the displacement DOFs:  
\[